I liked jazz in college and then pretty much hated it for a few decades. Now I've decided that, while almost nothing leaves me as cold as second-tier jazz and below, the A list contains a lot of artists so universal that I forego them at cost to myself.
Some people that got me back into it (apart from Coltrane and Mingus, whom I never really stopped liking):
Thelonius Monk: try Solo Monk or The Essential. Quirky and not especially moody, but so high-spirited and witty. If you get into his solo piano pieces, Bud Powell is an excellent next step.
Lester Young is totally classic and totally accessible: Pres and Teddy or With the Oscar Peterson Trio.
For the late-night, Taxi Driver mood: Coltrane, Ballads; Ben Webster, For Lovers; Ike Quebec, Blue and Sentimental; Jackie Gleason, Music, Martinis, and Memories. Quebec isn't really an A-lister, but this is a great album along these lines. The Jackie Gleason album would probably be filed under "Easy Listening," but you'll be surprised: the trumpet soloist is a real jazz guy, Bobby Hackett, and against a lush, swoony background orchestra, he really ventures into lonesome, moody territory.
You can also turn the clock back a few decades and find yourself drowning a torch in a swank film-noir nightspot with Ellington Indigos. No Duke Ellington fan's idea of his greatest, but gorgeous throughout.
Chet Baker: Chet Baker Sings.
Of course, if you want to go a little further out there's always Sun Ra. I know you said non-experimental, but an accessible way in is a compilation called Greatest Hits : Easy Listening For Intergalactic Travel.
