"middle of nowhere" is relative - huts and cabins are to be found within a couple hours' hike of a road, for reasons of economics. The truly remote spots have shelters at most. But this time of year there won't be crowds anywhere. Which means that many huts and cabins will be closed.
Off the top of my head (meaning, in New Hampshire): the Dartmouth Outing Club maintains some huts along the Appalachian Trail, the AMC has several in the White Mountains, and the Randolph Mountain Club has a few.
AMC also has cabins in Maine - come to think of it there are also cabins in Baxter State Park that stay open year-round. Those BSP cabins require a tedious hike in winter, when the road's closed, but otherwise they're probably not what you're looking for.
If you're looking for a cabin at some elevation, you're talking about RMC or AMC huts in the Whites. Most of the AMC huts just closed, but Carter Notch, Zealand, and Lonesome lake stay open year-round (self-service, nobody cooks for you). RMC's cabins are open year-round but don't take reservations.
There may be some cabins in Vermont along the Long Trail that I just don't know about.
If you are OK with the flatlands, you will have a huge range of choices - lots of hunting cabins for rent (though a little busy this time of year) or consider something non-traditional like this:
http://www.greenalpacayurts.com/PS if you're in CT you should also consider the Adirondacks - definitely some cabins there, not sure how many stay open though.