Odom began his career in sports at Goldsboro High School, in North Carolina, as the captain of his basketball and baseball teams.

After graduation in 1965, Odom accepted a job as the coach of Goldsboro High School while enrolled at East Carolina University in pursuit of a master's degree in physical education.

Odom stayed with the program until 1982, when he was offered a chance to return to the Atlantic Coast Conference (ACC) as an assistant at the University of Virginia.

As an assistant at Virginia (1982–1989) under head coach Terry Holland, Odom participated in some of the university's most successful seasons, including five National Collegiate Athletic Association berths and a trip to the Final Four in 1984.

Upon his arrival in Winston-Salem, Odom inherited a team that had suffered four straight losing seasons under head coach Bob Staak.

In 1995, led by Duncan, the team put up a 26–6 overall record, including winning the ACC tournament by defeating North Carolina by the score of 82–80, with the victory coming from a last second shot by point guard Randolph Childress.

During his tenure at South Carolina, his team earned one NCAA tournament bid, and made three appearances in the NIT, winning the championship in 2005 and 2006.