Not as well-known as some of his Philadelphia-bred contemporaries (which included drummer Philly Joe Jones, saxophonist John Coltrane, trumpeter Lee Morgan, and the Heath Brothers), Specs Wright nevertheless had a brief but distinguished career as a sideman, playing with some of the premier bop and hard bop musicians of his day. Wright played in a U.S. Army band before leaving the service in 1947; he then played in a group with saxophonist Jimmy Heath and another with trumpeter Howard McGhee. He joined trumpeter Dizzy Gillespie's big band in 1949 (Coltrane was also a member; Wright was the drummer on the band's November 1949 studio sessions, Coltrane's first commercial recordings). He stayed with Gillespie's big band until it folded in the spring of 1950. Later that year, he recorded with a Gillespie sextet that included Coltrane, Jimmy Heath, bassist Percy Heath, and vibist Milt Jackson. During the '50s, Wright played and recorded with such musicians as R&B saxophonist Earl Bostic (Coltrane was again a fellow bandmember), pianist Kenny Drew Sr., saxophonist Cannonball Adderley, drummer Art Blakey, and singer Carmen McRae, among others. He also freelanced in and around his hometown. In 1958, Wright played with saxophonist Hank Mobley's septet on the classic Monday Night at Birdland LPs (on the Roulette label) with Mobley, saxophonist Billy Root, trombonist Curtis Fuller, pianist Ray Bryant, bassist Tommy Bryant, and Lee Morgan. He also played with tenor saxophonist Sonny Rollins and vocalist Betty Carter around that time. In 1959, he replaced Art Taylor in Red Garland's trio for a short period; the band recorded with tenor saxophonist Coleman Hawkins during Wright's tenure. Wright also backed the vocal group Lambert, Hendricks, and Ross in 1960-1961.