I moved up from Kingsport, TN ~2 years ago, and really miss some of the beautiful spots down there. However, I'm not actually sure where Jeff City is. Of course, you should get the AT guide book and maps for the TN/NC section. The visitor center at Mt. Rogers and lots of local outfitters have good maps for that area (particularly Mt. Rogers Outfitters in Damascus, VA. They have a whole map room, and plenty of books on the subject). I think this time of year is a great time to get up to Mt. Rogers since it's so rugged looking, assuming you have some experience in that type of exposed area since several people each winter seem to get lost/stranded/rescued up on that ridge. There are shelters along the AT, but no numbered sites that I can think of. There are probably a few overused areas that they direct people away from at times, but not usually a big problem. Always good to check on Mt. Rogers/Grayson Highlands websites for updated details.
Some of my favorite areas include, Grayson Highlands(Wilburn Ridge Trail), Roan Mountain/Round Bald/Hump Mt./Yellow Mt., Laurel Creek Falls, Unaka Mt., etc. There are lots of balds and rhodo/hemlock shaded creeks to explore. Make sure to explore the town like Hot Springs, Damascus, Erwin with their generous hostel owners like Miss Janet & Bob Peoples.
Hot Springs, NC usually has Trailfest and Damascus,VA has Trail Days in the spring.