Must do's: Nebo-Hickory, it's close 10 miles away and you can easily get in a 50+ mile ride there. Nebo has a great flow and Hickory has some huge hills. Get the Mountain bike tourist map at the bike shops in Columbus or Nashville
Valley Branch Retreat 5 miles from the park entrance and 20+ miles of trail with a totally different feel than Brown County.
Gnawbone Camp is closed to riding for the next 5 weeks for the summer camp operations. Otherwise it would be tops on the list.
French Lick is great but a two hour drive 1 way. as Is Versailles.
If I had to choose I'd plan on 2-3 days riding at Brown County, A day at Valley Branch, a day or 2 in Nebo-Hickory , and then maybe a trip to French Lick or Versailles if you want a change. It's hard to imagine doing 4 hours round trip in a car after driving that far to ride, but french lick may be worth it.