To get to good "large area" singletrack trails from Smackover, I have the choice of traveling 95 miles west northwest to Bear Creek (Lake Greeson), 100 miles northwest to Bismarck (Ross Foundation Property), 220 miles north to Brock Creek, or up to 200 miles south to one of the areas in Louisiana. (Used to go 140 miles south to Breezy Hills, and hope to be able to ride there again one day soon.)
It's always been worth it for me to drive, however long, in order to take the family riding.
There is a private trail on Sturkey Road near Salem, where the Arkansas Hare Scrambles Championship Series holds an event. The owner might allow you to come over and bring the kids for a day of singletrack fun. You can get his contact information from the AHSCS schedule on their web site. Also, one of the Arkansas Dirt Riders lives near Batesville, and has a trail at his property. PM me to get his contact information.
Good Riding!
Jim