Hello, OP! I am also in the Greenville area and just this morning started down the road to getting good with a precision rifle (shot mine for the first time today). Anyway, a couple of very helpful gentlemen at the range pointed me to
the Coastal Carolina Rifle Club.
The club is quite a hike from Greenville (on board MCB Camp Lejeune in Jacksonville, NC), but they have almost monthly medium and long range competitions (600-1000 yards). I don't know if I'll make it this month, but I plan to be there with bells on in September.
Good luck, and if you decide you want to go to one of these, let me know and I'll buy you a beer up there.
ETA: I know it's not training as in a class, but it's certainly where you can regularly put what you learn to work in a class, which I guess is also training.