Mucci began as a baritone horn player and was appearing in professional settings by the time he was ten years old. As a teenager he switched to trumpet and worked in the late 1930s with Mildred Bailey and Red Norvo before joining Glenn Miller's ensemble in 1938-1939. During World War II he played in the bands of Bob Chester, Hal McIntyre, Claude Thornhill, and Benny Goodman. In the first half of the 1950s he worked as a house musician for CBS and also recorded with Buddy DeFranco and Artie Shaw; later in the decade he worked with Miles Davis, Helen Merrill, and John LaPorta. His association with Davis lasted into the early 1960s; he also played with Kenny Burrell in 1964.