 Conn eventually won his first medal in the 1954–55 season, when Hearts defeated Motherwell 4–2 in the 1954 Scottish League Cup Final.The following season, Conn scored as Hearts defeated Celtic 3–1 in the 1956 Scottish Cup Final, their first triumph in that competition for 50 years.He ended that 1955-56 season at the peak of his powers aged 29 with a career best 29 goals from 41 games.