I did a trip through Beijing Hikers a few years ago (I think someone in this forum recommended them):
https://www.beijinghikers.com/

Can't recommend them enough. Yes, you do have an insanely early start, but you need it to get out of the city and close to the wall. It was on a Saturday in November, and I think we only saw 2-3 other people (outside of security checking for permits). The dinner after was also great. Then after getting back into town, a bunch of us went out to the night market. Most of us slept on the bus both to/fr the wall. There were sections of the wall that were in pristine condition, and others that weren't (we actually had to hike off of it a few times, then back on). They keep the groups small (I think 15 or less), and have 2 guides (a lead and a chase).