Welcome to Memphis!
I live in east Memphis and can give you a list. Keeping it within an hour away is kinda tough, but I will list some further away places, too.
1). Chewalla Lake Recreational Area- (Mississippi) 50 minutes from East Memphis
2). Wall Doxey State Park - (Mississippi) 1hr from Memphis
3). Meeman-Shelby Forrest State Park (TN) - 30-40 minutes north of downtown Memphis
4). Chickasaw State Park (TN) - 1.5 hrs from East Memphis
5). Natchez Trace State Park (TN) - 1.75 hrs from east Memphis
6.) J P Coleman State Park (Iuka, MS/ Pickwick Lake TN) 2hrs from Memphis
7). Buffalo River (Loebeville, TN) 2hrs from Memphis
8). Reelfoot Lake State Park (TN) 2hrs 10 mins from Memphis
9). Mousetail Landing (TN River) 2hrs from east Memphis
10.) Fort Pillow State Park (TN) 1.5 hrs north of Memphis (small lake)
11.) Crowley's Ridge State Park (AR) 1.75 hrs from East Memphis
Google them for details or go to the state's Parks website for full details on what's available and rates.