Moncur was a multi-instrumentalist as a teenager, learning trombone, tuba, and double bass while growing up in Miami.
After moving to Newark, New Jersey, Moncur began playing bass on a local radio station, where John Hammond heard him.
Hammond brought Moncur in for studio sessions in 1935–36 with Mildred Bailey, Bunny Berigan, Putney Dandridge, Bud Freeman, and Teddy Wilson.
He was a founding member of the Savoy Sultans, playing with the group until 1945.
[1] His discography roster also showed he played sessions with baritone and bass saxophones in various 1930s and 1940s recordings.