While there is plenty of National Forest north of Hot Springs, Highway 7 north to Ola, AR. Just not a lot there, quite a lot of NF gravel roads and plennty of pull off spots to camp, just not my idea of a great camping place. If you want in the Ouachita's, better to be just east of Mena on the Cossitat (sp?) River. Around the Albert Pike Rec area (rec area now closed) Plenty of camping and quite nice.
Or go futher east to the Lake Sylvia area. You can actually take backroads (NF) from Hwy 7.
Or drive a couple of hours north to the Ozarks, Around Longpool NF campground, just noth of there is a great place on the Little Piney called "Treat" This area is, in my idea, much nicer than north of HS.
Feel free to PM me if you meed more detals..
Good luck and enjoy.