I strongly recommend getting the Ultimate US Campground finder app for your smartphone/tablet if you don't already have it. It will put you on some great stuff.
All the NP campgrounds are no hookup and limited services, plus they're covered up this time of year.
I recommend Indian Boundary in Tellico Plains, Tn. It's on the Cherohala Skyway and is a real gem. Cataloochie is supposed to be a really good one but I've never been, it's on the east side of the park off I-40 between Knoxville and Asheville. Both of these close at the end of Oct. though, I think.
Tennessee is full of beautiful State Parks. Just pick one. They all have full services and are open year round. You can reserve a lot of them on Reserve America.
I've stayed in a lot of the Georgia State Parks too, but they have weird rules like 2 night minimum stay on weekends and I think they have their own reservation system. If you're on the way up from Florida check out Cloudland Canyon. It's amazing but a little west of Chattanooga in the NW corner or Georgia and not really in the Smokies. It is at the edge of the mountains, kind of the back side of Lookout Mountain.
If you decide to get off the beaten path for some dispersed or primitive camping let me know, and I'll have a bunch more suggestions.
I live just south of Knoxville and primarily spend time on the West side and south sides of the NP. Someone from the Asheville area might have some other great suggestions for you on the east side, maybe Chris from Ujoint?
If you're OK with commercial campgrounds there are a few good ones in Townsend, Tn., said to be the 'quiet side of the Smokies', but I've never found the quiet side. For me, the quiet side of the Smokies is in Colorado somewhere. It's the most visited NP in the nation. Still well worth a visit though. If you can sightsee there during the week I would recommend it.
I'm glad to throw out some more if needed. Standing Indian in Franklin NC is amazing too, but not sure about hookups.
Currently vanless. Weird.