Looking for a place to play on the way to watch my alma mater (Lipscomb) play in the NCAA tourney for the first time. Go Bisons.
Willow Creek Golf Course is probably the best public track and it is not too far off I-40. Moderately priced. Tennessee National is semi-private but is a fantastic lay-out. You should be able to get on there if you call the shop. It is past Knoxville in a Loudon but is on the way to Nashville.
Agree with @zgosselin here. Willow Creek is going to be the best public track in the Knox area. Played it probably a dozen times and it doesn’t get old. If you get to Loudon, TN National is damn fun too (also heard Toqua GC in Loudon is good but never played it).
Windriver is pretty good as well. Layout is a bit funky and not walkable, but the course is usually in good shape and there are some cool holes.
Willow Creek is a solid option as others have said.
If you can get on Holston Hills do that instead.