Garner served as a political inspiration and mentor to Dolph Briscoe who was a member of the state legislature from 1949 until 1957. Briscoe was elected governor in 1972 and served through the oil-boom years of the 1970s, during which he increased spending for highway improvements, signed into law the Texas Open Records Act and streamlined state agencies. He was one of the state's leading ranchers and president of the Texas and Southwestern Cattle Raisers Association. In 2008, The University of Texas at Austin announced the naming of its Center for American History after Gov. Briscoe, in recognition of his support for preserving and promoting Texas and U.S. history.
The Briscoe Center plans to maintain the existing exhibit space devoted to Garner on the museum's first floor, and create new exhibits dedicated to Briscoe on the second floor, which previously had been closed to the public.