I did a short stretch of the Appalachian trail near the Chattooga river. That part of the country where North Carolina, South Carolina and Georgia come together is beautiful and home to the highest waterfalls in the Eastern US.
I'm more familiar with the Eastern half of the state than your neck of the woods but for shorter trails, I'd start by looking at Roaring River State Park or around Lake Taneycomo.
For longer trails, I'd go farther east like the Ozark trail near the Eleven Point river. The Bryant Creek/North Fork and the Jack's Fork/Current River areas are nice too. The Current river near Big Spring has some nice trails.
A little farther east, the Pickle Springs trail, though short, has some amazing geology.
Oh, and the St. Francois river area around Silver Mines is nice.
Or to the Arkansas Ozarks where the scenery is amazing. Or even the Oachita Mountains. I hiked there a bit many years ago.
Try the Mt. Sequoyah trail near Fayetteville or the Slaughter Pen trail near Bentonville.