Leishman started working with the New York Yankees' farm teams in 1947. He originally worked as a business executive for the Ventura Yankees before becoming their general manager. The same year, Leishman was named director of multiple Yankees affiliated teams including the Victoria Athletics and the Bisbee-Douglas Copper Kings.[5] He proceeded to the Salt Lake City Bees in 1950 as their owner before becoming the Bees' general manager when they relocated to the Pacific Coast League in 1957.[6]
Leishman stayed in the PCL as front-office boss of the San Diego Padres from 1960 to 1968, under owner C. Arnholdt Smith. When the Padres moved to the National League as an expansion team in 1969, Smith was awarded the MLB franchise. He took on Los Angeles Dodgers general manager Buzzie Bavasi as a minority owner, and named Bavasi president of the Padres. Although Bavasi was integrally involved in the team's baseball operations, Leishman was retained as general manager of the MLB Padres until his death in 1972.[7]
During his career, Leishman was twice honored as Minor League Executive of the Year by The Sporting News, in 1959 with Salt Lake City and 1964 with San Diego. He was named the minor leagues' King of Baseball in 1966.