There is 1) a leanto and tentsites at Horns Pond, near intersection of Horns Pond Trail and the AT, 2) a leanto in the col between West and Avery Peaks (I don't rtemember any tentsites there), near the intersection of Fire Wardens Tr and the AT, 3) some nice tentsites with privvies at the Fire Wardens Trailhead, 4) tentsites at Round Barn on Flagstaff Lake at the end of Safford Brook Tr a few miles off the AT, and 5) a leanto at Safford Notch just off the AT. These are all shown on the AMC Maine Guide maps of the Bigelpow Range. All are free and first come first served.
There are also three Maine Huts for more luxurious accomodations (state of the art off grid facility, bunkrooms, indoor plumbing, meals, etc.) all within hiking distance from the AT and all charge a fee like the AMC huts but, given the accommodations and quality of the food, I'd say worth the price. We spent part of last winter pondering permutations for a few days of hiking using some combination of huts, leantos, tentsites ...