Don't know if you have time to spend or want a quick overnight, but Mammoth Cave National Park is interesting. We camped at the main campground next to the visitor center in April and it was super nice because only about 10 campers were there. Could be a little crowded now that schools are letting out. But you can ride bikes to the visitor center and take a tour of the cave. Also have a great bike trail on the former railroad line right from that campground. There is a second more isolated campground in the less developed section of the park but we didn't have time to explore that section.
I second Mammoth Cave. Great place and not too far from where you'll be going through. There's also a pretty lake and campground on I-24 just north of Chattanooga. I can't remember the name. The interstate goes right through it, or over it. Might be a great place for an overnight if needed. Fills on weekends IIRC.
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