Sorry I don’t have specifics, but my son lived in Manhattan for the past year, carless, and found ways to head upstate using buses and trains and then from the last town to the trailhead, an Uber or Lyft.
Look at MegaBus (cheap, like $24, but mostly city to bigger city) and Peter Pan Bus Lines which isn’t as cheap ($45-ish one way) but goes to more small cities in southern New England.
Between them, you could get to Great Barrington MA, Pittsfield MA, Montpelier VT among many others, but those are particularly close to the AT / Long Trail.
There also must be meet-up groups out of NYC to hike upstate. Whether someone has or rents a car for the weekend, that would be an inexpensive way to get out and back. I’ve also seen that sort of group gel at a city youth hostel – someone proposes a trip and has or rents a car and invites others along to split expenses (although it’s usually to a more iconic (Grand Canyon) or urban destination than a backpacking trip.
If you’re college- or grad-student-age, look at university outing clubs (SUNY, Columbia, etc). Ours at UC Berkeley let non-students (but who were student-like) attend meetings and go on trips. Again, there’s a big cost savings in sharing the ride, plus there’s often a sharing of beta or leadership – someone knows a place a place to go while someone else scores the ride, and others source meals.
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