New defensive coordinator Brian VanGorder is expected to work with players at all positions on defense, but not be responsible for a specific position.
"After carefully evaluating our staff, I feel this restructuring best suits our program," Auburn head coach Gene Chizik said on Tuesday.
Martinez was officially hired by the Tigers on Friday, making the move from Oklahoma, where he spent the past two seasons coaching defensive backs.
For the past three years under Chizik, the Tigers used two coaches to handle the secondary. Phillip Lolley coached the cornerbacks and Thigpen coached the safeties.
Lolley is no longer an on-the-field coach, but is on the staff in an administration position.
Thigpen has an extensive background coaching linebackers. He coached the position for four seasons at North Carolina prior to coming to Auburn.
Thigpen also coached linebackers at Illinois in 2004. Prior to that he coached cornerbacks for the Illini.
An All-ACC linebacker at North Carolina, he got his coaching start there as a graduate assistant coach in 1998-99 after playing in NFL Europe. He coached linebackers at Tennessee State in 2000 and in 2001-2002 special teams coordinator and cornerbacks coach at Bowling Green.