Dunmore joined West Ham United during the 1959–60 season in an exchange deal that saw Johnny Smith move in the other direction. He made his West Ham debut on 19 March 1960 against Blackburn Rovers, and went on to make 39 appearances for the club, scoring 18 goals.
Dunmore joined West Ham United during the 1959–60 season in a trade that included Johnny Smith moving to another club. He made his debut for West Ham on 19 March 1960 against Blackburn Rovers and played 39 games for the team, scoring 18 goals.