If your at Limekilm, you also have the Moose River Recreation Area to hunt. There was an area outside of Old Forge back towards Carter Station that might be town land. (5-6 miles east then heading north on a gravel road) It used to belong to the owner of Old Forge hardware but was condemed after a lawsuit by a 3-wheeler accident. The land might be in town hands now. Lots of beech on the ridges back that way and you can walk in the old Adirondak Railway right-of-way for miles.
There are also areas owned by paper companies but I can't remember the name of the company. INTERNATION PAPER COMPANY!!!! Hey, the mind isn't gone yet!!!
They own lots of acres up there and lease hunting rights. If you can find them, contact them, they might have land not currently leased and send you a map of the lands. There office was south of Rochester if I remember correctly.