Let's see:
- Likely to be open.
- About 2-4 miles in
- Prefer a nearby body of water.
- Great ADK feel.
We will be coming from the Southeast.
I read the request differently as well, and thought the poster may have meant the southeast Adirondacks. In any case, I didn't assume High Peaks. I also assume that by "likely to be open" the poster meant "likely to be free". IF they meant southeast Adirondacks, there are several lean-tos in the Black Mountain region on the east side of Lake George. You can walk to Black Mt. Pond, Lapland Pond, and Milman Pond lean-tos all within 2-4 miles, and if you stretch it to five, include Greenland and Fish Brook Pond lean-tos as well (this last one has two lean-tos). All are right on a body of water. I'm not sure what a "Great ADK feel" is, but they're all beautiful short Adirondack hikes to me. Among all of them, Milman Pond is, IMO, the nicest, and probably the least used, in immaculate condition when I was last there, and an outhouse in pristine condition. The pond is small, with a short boardwalk leading out a short way into the pond, with a nice rock to fish from or read on. The other ponds are larger. Greenland is slightly more remote, and the path and leanto not used often (no outhouse here), but the nicest pond of the ones I've mentioned.
Of course, this doesn't even scratch the surface of possibilities.