Bunce established himself as a frequent member of the first team squad in 1958–59, making 26 appearances in all competitions. He followed this up with 32 appearances in 1959–60, as Watford secured promotion from the Fourth Division. After missing the first three games of the following season, Bunce scored his first career hat-trick, in a 6–1 win over Brentford at Vicarage Road on 30 August 1960. He played in all of Watford's remaining league games, finishing the season with 16 goals from 43 appearances. The club finished the season in 4th position, at the time their highest ever Football League placing. He followed this up with 11 goals from 51 appearances in 1961–62. After making seven appearances in the first four months of the following season, Bunce transferred to Cambridge United, and shortly afterwards also played for their rivals, Cambridge City.
