After completing his National Service, Simpson moved to Third Lanark in 1950. He then joined Newcastle United six months later for a fee of £8,750. He won the FA Cup twice with Newcastle, after taking over as their regular goalkeeper from Jack Fairbrother, in 1952 and 1955. After nine years and almost 300 appearances for Newcastle, Simpson returned to Scotland, with Hibernian, in 1960. His performances helped to save Hibs from relegation in 1962, but he fell out of favour under the management of Jock Stein due to his attitude in training. Hibernian sold Simpson to Celtic in 1964 for £4,000.