No.
It is only legal to shoot a wildlife species in Tennessee if there is a listed hunting season for that species in the hunting regs. Some species (coyotes for example) are listed with a year-round season, but it's still a listed season. No listing in the hunting regs means it's illegal to shoot the animal.
I guess there are narrow exceptions for self-defense and things like that, but I think in Tennessee, you're as likely to need to defend yourself against a sasquatch attack as you would against a mountain lion attack.
bd