Odom was born November 29, 1933 in Washington, Oklahoma; her parents were David Hiram Odom and Opal Lee Odom. The family moved to Dimmitt, Texas, in 1948. At Dimmitt High School she played four seasons of basketball, scoring 4,022 points and averaging 41 points per game. In 1951 she set the Texas single-game scoring record of 78 points. She led her high school team to state titles three times and was selected for Texas All-State honors each of those seasons. Odom helped lead the Dimmitt team to victory over the Iowa state champions twice, beating Slater High 47–20 in 1950 and Hansell High 39–30 in 1951.
Odom went on to play for the Wayland Baptist team in Plainview, Texas from 1952 through 1956. At the time the school played against women's Amateur Athletic Union teams, most of which were run as semi-professional teams and were sponsored by businesses. Wayland convinced local businessman Claude Hutcherson, who ran a passenger and cargo air service in town, to fly the team to its away games. The team was renamed "Hutcherson Flying Queens" and became one of the most successful AAU women's teams. During her time at Wayland the team went 115–5, and began a series of 131 consecutive wins which ended in 1958. This 131 game run remains the longest winning streak in college and professional basketball history. Odom was the team's top offensive player all four years, and she was the first Wayland player to become a four-time All-American. The team won three AAU titles while she was at Wayland, and she was the 1956 AAU Tournament MVP. Two of her years Wayland went undefeated. While she played, Wayland teams set several school records which still stand including season wins, consecutive wins, and best defensive average. Former Wayland coach Harley Redin said of Odom, "She was such a great post player. Her fundamentals were just perfect. She was just exceptional at faking and getting good, accurate shots." Redin imposed some rules on his team, including that the players could not be married.. When Lometa's boyfriend in Dimmitt, Eloy Powell, proposed to her, she told him they had to wait. Just a few weeks later, Powell was killed by a train returning home from a visit to the campus. In 2013 Odom said, "I thought I’d never get over it, losing him. Some days I just wanted to go back to the farm. But I stayed. I knew I couldn’t let the Flying Queens or my daddy down." Odom was chosen for the women's basketball team to represent the U.S. at the Second Pan American Games in March 1955 in Mexico City. The U.S. team went 8–0 and captured the gold medal.