Mumme's coaching career began as the offensive coordinator at Foy H. Moody High School in Corpus Christi, Texas from 1976 through 1978.
During his time as a high school and college assistant coach Mumme developed an unorthodox, pass-oriented offensive attack that proved very successful at moving the ball, gaining yardage and scoring points.
The unusual attack, utilizing short passes to multiple receivers and backs out of the backfield, allowed Mumme's teams to compete against more talented and athletic opponents.
While finishing 7–4 in his first season, he eventually led the team to the national quarterfinals in 1991, the first playoff appearance in the school's history.
In 1994 Valdosta State defeated the University of Central Florida 31–14, an upset over the team picked by Sports Illustrated in the preseason to win the NCAA Division I-AA national football championship.
Quarterback Chris Hatcher won the Harlon Hill Award as player of the year in NCAA Division II football.
On December 2, 1996, the University of Kentucky announced that it had hired Mumme to replace Bill Curry as head coach of its football program.
[7] Led by sophomore quarterback Tim Couch, Kentucky's offense set multiple school, SEC and NCAA records.
[8] In 1998 Kentucky won its season opener against Louisville at the christening of Papa John's Cardinal Stadium by the lopsided score of 68–34.
[3][4][12] A close loss to Georgia on a missed last-second field goal was followed by victories over eventual SEC West champion Mississippi State and Vanderbilt.
[13] Couch elected to forgo his senior season and enter the 1999 NFL draft, where the Cleveland Browns selected him with the #1 overall pick.
Press reports detailing a growing investigation into NCAA rules violations cut the legs out from under the team, and it lost its final eight games.
[3][4] In early 2001, Mumme resigned amid numerous NCAA rules violations, largely payments to recruits.
Bassett had been forced to resign in 2000 for giving gifts to prospects and writing papers for them, and was slapped with an eight-year show-cause penalty, which effectively blackballed him from the collegiate ranks until 2009.
While the committee concluded that Mumme “failed to monitor the activities of the recruiting office”, he was innocent of any rule violations[17] and he was not given any individual sanctions.
[19] In December 2004 Mumme was named the head coach at Division I-A New Mexico State University, replacing Tony Samuel.
McMurry won their first-ever playoff game against Trinity 25–16 before losing in the second round to ASC rival University of Mary Hardin-Baylor.
[27] In April 2021, Mumme was announced as the head coach of the Linemen,[28] competing in The Spring League North Division in Indianapolis.