It’s been a while since I lived there and you didn’t specify what area of the state you want to hunt, but 3 areas that have a lot of public land are:
Allegany County, the southern half of Cattaraugus County and the southwest quarter of Steuben County. Rabbit, Turkey, Pheasant, Deer & Bear.
The Adirondack Park - a few million acres. Snowshoes, Rabbit, Grouse, Deer & Bear.
Draw a line SW from Penn Yan through Naples to Wayland, then SE over to Hammondsport, then back to Penn Yan. A lot of public land in that triangle. Pheasant, Turkey, Rabbit, Deer.
I don’t know how good the DEC is anymore, but they might be a source for pinpointing the exact location of public lands for you. They used to be real helpful. Another source of information can be the county courthouse. The auditor’s office has a record of land ownership and, if you find someone helpful, can be a real good source.
One more suggestion - log on to www.nydeer.com,
go to their talk forums and ask the same question in their open forum. They seem to be mostly deer & waterfowl hunters, but might give you a lead or two. Most of them are situated in an area east of Rochester, or at least the real talky ones are. Almost zero public land there, but they got to get out of the area once in a while.