If you are in NYC why not explore your home state a bit, the Adirondacks are an amazing area. The park is huge and there's wilderness aplenty, some easier hikes and some pretty difficult ones as well.
The Catskills are also a very nice area, and closer too. The mountains are a bit smaller, and the park less vast so it could provide some easier hiking. I can't speak too much about campgrounds, but I'm pretty sure both parks at least have a few.
I'm from CT, and while there's even some pretty nice hiking on the AT in Conn and Mass, I do most of my serious hiking trips in the White Mountains in NH. I think anyone will agree that they're phenomenal.
I think you're in a pretty decent location, I mean I don't know much about hiking in the southern states, but you are in close relatively close proximity to some really nice ranges in New York and New England.