Moore graduated from West End High School in 1957 and played for Bear Bryant's Alabama Crimson Tide football team until graduating in 1960. From 1962 to 1964 he was the offensive line coach for Charlie Bradshaw's Kentucky Wildcats. In 1965 Gene Stallings hired him as offensive coordinator at Texas A&M University. In 1971 he went to work for Bill Dooley coaching offensive line for the University of North Carolina Tar Heels.
Moore came back to Tuscaloosa in 1972 and became Alabama's first offensive coordinator when the position was created in 1974. In 1975 he was hired to take over from Don Fambrough as head coach of the Kansas Jayhawks. He was named Big Eight Coach of the Years after his first season, which was capped by a 23-3 upset of eventual national champion Oklahoma. Over three seasons his teams had an overall record of 18-26-1. Fambrough returned as the Jayhawks head coach in 1979.
After retiring from coaching, Moore entered private business. In 1994 he was honored on National Philanthropy Day by the West Florida chapter of the Association of Fundraising Professionals. He has been active in bird dog field trials and showing Tennessee walking horses, having raised and owned multiple national champions in both venues. In 2010 he and his children, Mike and Kim, purchased the Dixie Nissan dealership in Bessemer and renamed it Moore Nissan.
- "Bud Moore (American football)" (January 22, 2014) Wikipedia - accessed January 30, 2014