A lean, righthanded starter for the A’s in both Kansas City and Oakland, the highly-touted Odom made his ML debut only three months after graduating high school in 1964, but did not reach the ML for good until 1966. A’s owner Charlie Finley ordered the 19-year-old Odom into the starting rotation at the end of 1964, and he made five starts, pitching one shutout but lasting only eight innings total in the other four, and recording a 10.06 ERA. Odom spent most of 1965 and parts of 1966 and 1967 in the minors, but made the AL All-Star team in 1968, finishing 16-10 with a 2.45 ERA. Odom was 15-6, 2.92 as Oakland’s third starter in 1969, and made the All-Star team for the second consecutive year, but was shelled for five runs while retiring only one batter in the All-Star Game.
Elbow problems began to hamper Odom at the end of 1969 and throughout 1970, then after an off-season in 1971 he was 15-6 again in 1972, helping Oakland to the first of three consecutive World Championships. He was dazzling in the LCS, shutting out the Tigers on three hits in Game Two and allowing a single unearned run in the deciding Game Five, and in the WS he lost Game Three to the Reds 1-0 and allowed only one run while getting no decision in Game Seven. Odom slumped to 5-12, 4.49 in 1973, and spent most of 1974 in the bullpen before being traded to the Indians with cash for Dick Bosman and Jim Perry in 1975, then was traded to the Braves for Roric Harrison three weeks later. His ERA was 7.22 in 1975, and he spent his final ML season with the White Sox in 1976, winning only two of eight starts. One, however, was against the A’s on July 28, when he combined with reliever Francisco Barrios to pitch a no-hitter.