Bama fans are split. Plenty of them, especially those beyond a certain age, point at Tennessee and the Third Saturday in October. Others, who tend to be a bit younger, look first to the Iron Bowl and Auburn.
There was a time when Vandy was our #1 rival. They were every bit as good as us back then (talking the first 20-30 years we played football). General Neyland flipped the script and we started beating them regularly, and it became too unbalanced to remain much of a rivalry. But it was there at one point. I'm pleased as punch we ruined it by beating them by winning most years.
Kentucky was a hot rivalry for a while, too. We had a trophy and everything (the beer barrel...though for the life of me, I don't understand why smart folks never switched it to a whiskey-bourbon barrel). But then that became too lopsided as well, the rivalry cooled down, and the trophy was retired after a couple of students died in an alcohol-related car accident.
Florida and Georgia are new things. Just since the early 90s have we played them annually. They are, to an extent, manufactured rivalries that will die back down once the SEC does away with divisions (just a couple of years from now).
North Carolina would be a natural rival, and a good one. But they have enough in-state rivals (NC State in football, Duke in basketball), plus being in different conferences it wouldn't easily work.
Virginia Tech would be a decent cross-border rival, as well. But same problems as for North Carolina.
Funny, we're one of only two states in the country that borders 8 other states, plus we have a fellow in-state Power 5 program (even if just barely), and all our rivalries tend to come with an asterisk. Which is why you started this thread.
But I'm perfectly fine with having a lot of rivalries that aren't extreme, rather than one or two that are red hot.
I like where we are, and our history.
Go Vols!