Hubbard started playing the mellophone and trumpet in his school band at Arsenal Technical High School in Indianapolis, Indiana. Trumpeter Lee Katzman, former sideman with Stan Kenton, recommended that he begin studying at the Arthur Jordan Conservatory of Music (now the Jordan College of the Arts at Butler University) with Max Woodbury, the principal trumpeter of the Indianapolis Symphony Orchestra. In his teens Hubbard worked locally with brothers Wes and Monk Montgomery and worked with bassist Larry Ridley and saxophonist James Spaulding. In 1958, at the age of 20, he moved to New York, and began playing with some of the best jazz players of the era, including Philly Joe Jones, Sonny Rollins, Slide Hampton, Eric Dolphy, J. J. Johnson, and Quincy Jones. On 19 June 1960 Hubbard made his first record as a leader, Open Sesame at the beginning of his contract with Blue Note Records, with saxophonist Tina Brooks, pianist McCoy Tyner, bassist Sam Jones, and drummer Clifford Jarvis. Six days later he returned the favor to Brooks, and recorded with him on True Blue.