Almost correct. Moore Hall was added to Hill Hall in 1955.
Hill Hall was built in 1939 and housed athletes (potentially including Tom Landry)
https://livingnewdeal.org/sites/university-texas-austin-hill-hall-austin-tx/
University of Texas at Austin: Hill Hall – Austin TX
City: Austin, TX
Site Type: Education and Health, Colleges and Universities
New Deal Agencies: Public Works Funding, Public Works Administration (PWA)
Completed: 1939
Designer: Paul Philippe Cret
Beginning in 1939, the university housed its athletes in Hill Hall. The five story building (including basement) is of Spanish Renaissance style with red roof and tan bricks. The Public Works Administration provided a grant of $46,636 and a loan of $57,000 to cover the cost of construction. When originally opened the building housed 84 men.
Hill Hall is named for Dr. Homer Barksdale Hill of Austin who volunteered to treat UT athletes from 1893 until his death on July 18, 1923.
As its athletic programs grew, the university added Moore Hall in 1955, just south of Hill Hall and connected to it, to house additional male athletes. Now known as Moore-Hill Residence Hall, it is open to the entire student body. The hall went coed in 2005.