Time And Again by Jack Finney.
Simon ("Si") Morley is in a bit of a rut. Dissatisfied with his job as a commercial artist (mostly for advertising agencies) in 1960s New York, plenty of interest from the ladies (hey, he's a good-looking guy) but trouble locating that special lady, you know, The One, he's ripe for a bit of a change and the whiff of adventure.
So when he's approached at his office by Ruben Prien, a similarly young-ish man with a shared US Army background, who tells Si that he really has to speak to him about something really really important - he can't tell him the details yet but, honestly, it may just be the most exciting thing ever - Si is inclined to give him the benefit of the doubt and turn up a couple of days later at one of those ostentatiously grey warehouses housing a suspiciously un-busy removals business that's clearly a front for a massive secret government project if you just slip the right secret access code to the receptionist.
Inside the seemingly endless series of office rooms and giant plywood reconstructions of various real-world locations Si meets the architects of the project, including Ruben, Dr. Danziger, the project's instigator and director, and Oscar Rossoff, a medical doctor who tests Si's susceptibility to, and reaction to, hypnosis. It's only after Si has passed this test with flying colours that the nature of the project is revealed: Danziger's research suggests that those particularly adept at a particular form of self-hypnosis can actually travel through time. The best subjects are those who have a particularly acute visual sense, something Si's artistic background is ideal for, as the government well knows - Ruben didn't just come knocking on his door at random.
Various selected operatives are being sent to various locations at various times to test out the limits of the technique - one seems to be that the specific location where the time-slipping person does their thing must be substantially the same in both times. Si's assignment is in New York in the early 1880s, and it just so happens that the Dakota building exists in both timelines and the government can pull a few strings to rent an apartment that was also known to have been empty in early 1882.
Si has chosen this date specifically because his sort-of-girlfriend, Kate, has a family mystery to solve involving a man called Andrew Carmody, whose son was her adoptive father. He was in possession of some mysterious letters including Andrew Carmody's suicide note, itself written at the bottom of an earlier letter apparently from someone else and arranging a meeting in City Hall Park on a specific date in early 1882. Why not, says Si, add a bit of focus to the jaunt back in time by solving this mystery?
Si takes up residence in the Dakota and, after a bit of living as an 1880s gent would - getting the feel of the clothing, growing a beard, giving up watching the TV and making phone calls - he is ready to enter the trance-like state required to transcend the apparent laws of the universe and slip back to 1882. After a couple of brief-ish early jaunts to test the water he finds that he can make the transition without too much difficulty and pops down to City Hall Park on the night in question to hide behind a convenient pillar and eavesdrop. Sure enough Carmody turns up, and the man he meets with turns out to have blackmail on his mind, Carmody having done some shady property dealings in the past. The blackmailer turns out to be a man called Jake Pickering, and Si, determined to get to the bottom of things, tails him to his lodgings and, on finding a free room available for rent in the house, takes it.
The family running the house includes a young woman, Julia Charbonneau, who catches Si's eye; she also turns out to be Jake Pickering's girlfriend. Si convinces her that he is a private detective and takes her along to stake out the office where Carmody and Pickering have arranged to meet to agree the handover of the cash. They hide in a boarded-up room next door (where a lift-shaft is being constructed) and await the outcome of the rendezvous. It turns out Carmody has no intention of handing over the sum Pickering demanded, ONE MILLION DOLLARS, and instead intends to fob Pickering off with ten grand. Needless to say Pickering isn't having this and an altercation occurs, during the course of which Carmody, in a panic, attempts to set fire to Pickering's filing cabinet where various incriminating documents are stored. Aaaaanyhoo, one thing leads to another, the whole freakin' building catches fire, Si and Julia are obliged to reveal themselves and flee to safety, and the whole building burns down with some loss of life.
No biggie, Si thinks, the loss of life aside - but, if you think about it, all those people were already dead anyway as far as Si was concerned, from his 1960s perspective anyway. So it's something of a surprise when he and Julia are arrested by New York's police chief, Byrnes, and charged with murder. Even more of a surprise when he then lets them both go, but it turns out he just wants to save on the inconvenience of a trial and shoot them while "escaping". Si and Julia avoid this immediate fate, but the net is closing in, and the only way Si can see for them to avoid being caught is to jump forward in time to Si's "present".They do this, which buys them enough time for Julia to be amazed at the brazenness of 1960s womenswear and for her and Si to realise that they're in love with each other, but Julia decides, as intriguing as this future is, she can't stay - besides, she's figured out what has happened with Pickering and Carmody and can fix things if she goes back.
Si reluctantly says goodbye and heads off to report to Ruben and Danziger for a debrief. It turns out that Danziger is no longer in charge, though - haunted, throughout the project, by the thought of one of the operatives doing something in the past which affects the future, from stepping on a bug to, I dunno, facilitating the destruction by fire of an entire building. Even Si's clumsy interventions don't seem to have had much effect, however, and in any case there is now a different school of thought, led by Ruben, which says: hell yeah, if we can change some shit, we should totally do it. but only by making bad stuff good, right, because that can never ever backfire.
Si agrees to go back and do his best to make the changes that Ruben suggests, but, having spoken to Danziger first, in fact goes back and makes a small but crucial intervention that ensures that Danziger's parents never meet, and therefore that the project will never get started. His conscience clear, he then strolls off into 1882 New York to locate Julia and start a new life with her.
I can't remember where I saw a recommendation for Time And Again, but I think it was in some "forgotten classics" list, and indeed the edition I own is in the Gollancz Fantasy Masterworks series, a sort of companion to the SF Masterworks series that has featured here a few times before - a non-exhaustive and non-chronological list goes: Roadside Picnic, I Am Legend, Last And First Men, The Sirens Of Titan, The Affirmation, The Dispossessed. An interesting contrast, and one revealing of my reading habits and genre preferences, is that Time And Again is the only novel on that Fantasy Masterworks page that I've ever read, though there are a couple of authors (M John Harrison, Ray Bradbury) who I have read other novels by. The fantasy genre in its most stereotypical form (castles, swords, quests, Things of Power, prophecies) has never really appealed to me, but of course there are boundaries where fantasy bleeds into other genres, most obviously science fiction. I'd say Time And Again straddles that boundary - it is mercifully free of dragons and goblins, and the central premise, i.e. time travel, is a science fiction trope, though it's described in exceptionally woolly terms here - you just have to really want to and think really hard and poof, apparently. None of the associated questions are really grappled with - what happens when you transition from the 1960s to 1882? Do you just pop into existence? What if someone else is standing in that spot? Given that the poofing from one timeline to another doesn't seem to involve any special equipment, why has it taken until now, and a multi-million-dollar government project, to work out how to do it? In fact, how come people aren't doing it accidentally all the time?
That doesn't really matter, though - even though we don't land properly in 1882 for any length of time until page 115 of a 400-page book, the bit in the "present" feels like a flimsy framing device to a lovingly-detailed historical description of late-19th-century New York. And a fascinatingly realised picture it is too - lots of stuff not in existence yet (the Empire State Building, most of the major bridges apart from the in-progress Brooklyn Bridge), a few things in unfamiliar situations (the Statue of Liberty's arm, temporarily housed in Madison Square Park, plays a key role) and a few things jarringly present that no longer are.
So it's loosely science fiction, arguably fantasy, certainly a historical mystery with a dash of romance thrown in. Its main fault is its main character's love for the historical period in which he finds himself. This sounds like an odd criticism, so stay with me: Si Morley makes no secret for his preference for the 1880s over the 1960s; well, fair enough, each to their own, but the main thrust of his criticism of the modern era seems to be that yes, OK, sanitation, medicine, the roads, but there are an awful lot of uppity black people and unattractively assertive women around, and he is deeply unimpressed with modern achievements like landing on the moon and inconsequential shit like that. In short, though he is a resourceful protagonist, he seems to basically be a bit of a jerk.
It hardly seems fair to criticise a book about time travel for some plot implausibilities, so I'll restrict myself to a couple: it seems unlikely that Julia could pop back into existence in 1882, even though she's seen through the fiendish plot that got them arrested, and have time to explain to the relevant people that she's onto them before being delivered a hot lead sandwich from Byrnes' boys. And even in pre-electronic-records 1882 it seems like Si Morley might have some explaining to do, before taking up his place in society, marrying Julia and all the happy-ever-after stuff, at his having apparently not existed anywhere before five minutes ago.
The other thing to mention about Time And Again is that it has illustrations: some purportedly Si Morley's own artwork, some photographs, mainly the supposed handiwork of one of the other tenants of the Gramercy Park boarding-house. Like Sweet Caress, which made use of a similar device, there is an occasional suspicion that some portions of the narrative only exist to allow the inclusion of an arresting image, like the one of the Brooklyn Bridge on page 258 - I know we're in pre-health-and-safety-assessment times, but it seems unlikely that a random member of the public could just swan up and walk out onto the main cables.
Anyway, for all that, Time And Again is great fun, and ties up its time-travel loose ends in a fairly satisfying way, as long as you don't think about it too much. Jack Finney's best-known work is probably his 1955 novel The Body Snatchers, mostly through its several film adaptations, as previously mentioned here.