The Valley Way trail is a direct and easy route to the Madison Spring Hut, but if you don't have reservations or the money for a reservation, the Valley Way Tentsite may be a suitable alternative. Fallsway is a trail that runs parallel to the lower part of Valley Way and the brook, giving you a great view of small falls.
3.1 miles from the Appalachia Trailhead you'll see a sign marking the spur path to the tentsite. There's a brief uphill and then you'll pass a spring on the left, a bearline, and then the tentsites and composting toilet. The map at the tentsite mentioned 6 sites, but I could only locate 2 tent pads and 1 fire pit; now that I'm home, I see the website also only mentions 2. I had friends waiting, so I didn't do a detailed search.
It's a quiet, sheltered spot below the treeline. It's your only tenting opportunity in the area.