Rice
Not too many teams could claim losses to Rice in the College Football Playoff era, with the Owls descending to the bottom of the standings and becoming a doormat in the lowly Conference USA, but there was a time when Rice was a powerhouse in the sport.
Of course, that was in the first half of the 20th century, so you’re forgiven if that sounds like a ludicrous statement. From 1937-1960, the Owls were part of six New Year’s Six bowl games, and they won four of them. In that 1937 season, Rice knocked off Auburn by six, and in 1938, the Owls shut out the Tigers. Both games took place in Houston.
Like SMU, Rice being in Texas makes it unlikely that Auburn football will look to have them on their schedule anytime soon, with in-state programs more of a priority for the non-conference early portion of the schedule.