With just one free day Id head to Central Texas around Austin the state capital. Have you made it to the capital yet? If you have time for museums the Bullock Texas State History Museum or the LBJ Presidential Library are really good and give you a solid feel for Texas history without being boring.
For fun stuff Austin has that live music energy downtown or on Rainey Street. Or head an hour west to Fredericksburg a historic German town with bakeries wineries pretty Hill Country drives and even a cool pioneer museum. Youll probably feel right at home. The Circuit of the Americas race track near Austin is another fun one even without a race going on you can tour it or just enjoy the views.
What you absolutely cant leave without doing is eating some good BBQ taking a scenic drive through the hills and soaking up that big open Texas feel. It really hits different once you get outside the cities.