It's a bit of a haul from Paris (I'm originally from Humboldt) but there is MSSA in Memphis. There range goes out to 600 yds. and they put on F class and HP matches among other stuff.
Have you tried Montgomery Co Shooting Sports complex? They have a 400 yard rifle range. It just south of Clarksville.
If you are active duty and retired Military, Ft Campbell's range 16 is a great place to shoot. Its open weekends and most holidays. You can shoot paper out to 200m and iron maidens out to 750 or so.
Someone was telling me that Range 16 at Ft. Campbell was open to the public on wkends now. Is this true? I know before it wasnt or ever but someone said they started to let them use it now.
+1 on MSSA. It's expensive to begin a membership there, not as much for the yearly fee. But I know that anyone can shoot the benchrest matches for like 18 bucks. Not sure about F-class but I would hazard to guess its the same way. PM me if you need some info, I can put you in touch with some good peoples.
Actually took the trip up to Southside TN to Montgomery Shooting Complex. Sweet place to shoot...only down side is there is 2 rifle ranges...one for siting in at 100yd and the other only goes to 400yd but there is no 100yd on that range..so you have to switch ranges after site in. But they do have a golf cart to take targets out to 200,300 and 400yds. The other cool thing is they have several ranges for every discipline. Pay your 5.50 for your range card get your 2 free targets and shoot.
Henry County Gun Club in Puryer TN has ranges out to 600. It's a great club with great facilities. It is not a public range though, If you would like to see it just pm me and I can meet you there.
MSSA here in Memphis usually has High Power, Garand, F-Class, BR, 3 Gun, Trap, Skeet and Cowboy going on just about all month every month.
600Y available for matches.