Smith was the fifth of six children of Oury and Ruth Smith of Farmington, Kentucky. The family lived in a farmhouse that had no electricity and no indoor plumbing. He was nicknamed "Odie" after a comedian on the Grand Ole Opry. As a child, he attended a three-room schoolhouse in rural Graves County, Kentucky. Because the family didn't have money for a basketball, he learned to shoot one his mother made from rolling up his dad's socks. He attended Farmington High School, where he nearly didn't play high school basketball until the school's principal/basketball coach agreed to give him a ride home (a distance of seven miles) after practices. As a senior, his only scholarship offer was from nearby Murray State University, but he took too long to accept and the offer was withdrawn.

Smith enrolled to play basketball at Northeast Mississippi Junior College (now known as Northeast Mississippi Community College). After Smith excelled on the court, Northeast coach Bonner Arnold convinced legendary University of Kentucky coach Adolph Rupp to send a scout to see Smith, and UK offered a scholarship.

Smith didn't see much action his junior season until Kentucky's star guard, Vernon Hatton, went out with an appendectomy, and for seven games, Smith averaged 16.3 points. As a senior, Smith was a starter and averaged 12.4 points per game. The Wildcats' team, known as the "Fiddlin' Five", beat Seattle University 84-72 to win the 1957–58 national championship, led by 30 points from Hatton and 24 from Johnny Cox. Smith averaged just under 14 points in UK's four NCAA tourney wins, including seven in the championship game.

Smith graduated from Kentucky with a business degree.

He was selected in the 1958 NBA draft, but not until the 15th round (85th overall) by the Cincinnati Royals. Instead of attempting to make the Royals, he joined the U.S. Army, where he played on the Army's all-star team and in 1960 was selected to play on the eventual undefeated U.S. men's basketball team that won the gold medal in the 1960 Olympics in Rome, Italy. The team went 8-0 in the Olympics, led by future hall-of-famers Oscar Robertson, Jerry Lucas, and Jerry West, although in the opening win against host Italy, Smith was the team's co-leading scorer as Smith and Robertson each totaled 16 points. The team's average margin of victory in the eight games was 42.4 points per game.

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