Reece was born January 5, 1931 in Kingston, Jamaica, the son of a silent film pianist. He attended the Alpha Boys School (famed in Jamaica for its musical alumni), switching from baritone to trumpet at 14. A full-time musician from age 16, he moved to London in 1948 and spent the 1950s working in Europe, much of that time in Paris. He played with Don Byas, Kenny Clarke, Frank Foster and Thad Jones, among others.
Winning praise from the likes of Miles Davis and Sonny Rollins, he emigrated to New York City in 1959, but found New York in the 1960s a struggle. Reece recorded a series of critically acclaimed records on the Blue Note label, which were reissued on Mosaic in 2004 that gave fans hope of a comeback. Still active as a musician and writer, Reece has recorded over the years with Victor Feldman, Tubby Hayes, Paris Reunion Band, Clifford Jordan's Big Band, tenor saxophonist Dexter Gordon, fellow trumpeter Ted Curson, pianist Duke Jordan, long-time Sun Ra alumni John Gilmore and drummer Philly Joe Jones.