This is a bit of a puzzle. Was this a particularly large (like, hand-sized) spider? From everything I’ve looked at it really seems like this spider matches up with African Trapdoor Spiders, but it seems strange that one would just be wandering Tennessee when, from everything I’ve seen, they are definitely not native to that area. I’m a bit out of my wheelhouse here but that really does look like an African Trapdoor Spider, almost uncannily so. Someone else who lives in the eastern US could probably provide better information than I could, but that’s my best guess.
This is a spider in the Antrodiaetus genus. They are referred to as folding-door spiders. In North America there are 8 families of mygalomorphs including the Theraphosidae (Tarantula) family. The rest are trap-door spiders. It is safe to say there are hundreds of species in the Mygalomorph infraorder native to North America. The only species I found in this genus for Tennessee was Antrodiaetus unicolor so it might be that. The green mark on its back could be a parasitic wasp or fly larvae but its too hard to tell.
Thank you for the further information! Smaller orb weavers and jumping spiders are more of my specialty. Your identification seems right to me, though I was wondering about the green patch myself.