Carnett was noticed by big league scouts when he was pitching in high school. He was signed by the Chicago Cubs in 1935, assigned to the Class C Ponca City Angels of the Western Association in his hometown, Ponca City, Oklahoma, and won 19 games. In 1936, he was in spring training with the Los Angeles Angels, the Cubs' top farm club in the Pacific Coast League, but an accident during a "pepper" game led to a serious shoulder injury that jeopardized his career. He was reassigned back to Ponca City, pitched with the injury and lost ten straight games, took off some time to work on his hitting, and then won 16 consecutive games.