In '38, UT was named National Champion by Billinglsy, Boand, Dunkel, Football Research, Houlgate, Litkenhouse, Poling, and Sagarin (at least Sagarin was awarded retroactively.) TCU won the AP (which is, along with the later introduced coaches poll, the NC most recognize as legitimate.) TCU was 11-0 and SWC conference champions that year. With apologies to Villanova and Georgetown, UT and TCU were the only two major football schools to go undefeated. Tennessee, ranked #2, beat Oklahoma in the Orange Bowl 17-0. TCU went on to beat Carnegie Melon in the Sugar Bowl 15-7.
In '39, Texas A&M was a near consensus NC (including the AP,) with twice-tied USC winning Dickenson, and Cornell winning Litkenhouse and Billingsley. Tennessee was undefeated going into the Rose Bowl with a 23 game win streak, but lost to USC 14-0. UT star George Cafego did not play in that game due to injury. A&M would go to the Sugar Bowl and defeat Tulane 14-13.
As mentioned earlier, the NC was awarded before the bowl games, but I thought the results were interesting enough to mention. In both seasons, however, there was another undefeated, untied team from a major conference that got the nod over UT.