From Austin you have plenty of hiking in the Hill Country. With driving there is Big Bend, Davis Mnts or Guadalupe Nat Park. Weather could be glorious from Oct onwards.
Looks like there is also some availability to PHX early next year. Depending on weather there could be Grand Canyon (South Rim), around Sedona, Chiricahua Nat Mon, Organ Pipe Cactus Nat Mon (perfect in the Spring)........ It could be freezing with snow of course.