The first thing is; do you desire a certain snow, possible snow or certain no-snow? If you want relatively snow-free, then try something not too high in the SouthWest. There are many choices in NM & AZ, try the National Forests or BLM lands. In particular the various Congressionally-designated Wilderness areas.
If you want certain snow, then you could go to the NorthWest, King's Peak in UT had snow up high in the 3rd week of Sept. The options would be ID, MT, WA, WY, CO, UT, and maybe parts of OR. As before, check the BLM & USFS lands.
Possible snow could be had in any number of places, but close to home is upstate NY, VT, NH & ME. There is public land and trails in those states, including the AT & Long Trail.