Is there another P5 school besides Tennessee that does not have a clear cut "biggest rival?" Not only that, but the schools that we consider big rivals (Florida, Georgia, Alabama) do not consider Tennessee to be a big rival. Also, we do not consider the schools that consider us to be their biggest rival (Vandy, Kentucky) to be a big rival.
We're kind of stuck in no-man's land as far as rivalry games. We consider Florida, Georgia, and Alabama to be huge rivalry games but they do not consider us to be their biggest, or even #2 rival. And we do not consider Kentucky and Vandy to be big rivalry games, even though they consider them to be.