He immediately became a regular in the side, winning the Scottish Football League in his first season and the Scottish Cup in his second while also being involved in the semi-finals of the 1959–60 European Cup[2] and earning selection for the Scottish Football League XI,[3] but then lost his first team place after Jim Baxter joined Rangers in 1960; Baxter initially played at inside forward before being moved to Stevenson's left half berth.

He made his debut in the 2–1 home league defeat to Burnley on 3 November 1962, he scored his first goal in a 5–0 victory over Leyton Orient at Anfield, Stevenson managed to hit the net in the 35th minute, Roger Hunt with a hat-trick and Ian St John scored the others.

Stevenson missed just the one fixture and scored five times as Liverpool topped the table by a comfortable 6 points over Leeds.

[4] Stoke manager Tony Waddington paid £30,000 for Stevenson as a replacement for the departed Calvin Palmer.

[1] In 1974, he joined the Vancouver Whitecaps in the newly formed North American Soccer League (NASL).