Originally thought of as a potential replacement for either Jack Charlton or Norman Hunter as the Don Revie Leeds team started to break up.
Stevenson had to wait until April 1976 before he got anything like a run of games when he made six successive league appearances replacing Paul Reaney at right back.
He joined Bristol Rovers in 1985, but decided to retire the following summer aged 29, after struggling with injuries.
Following his retirement from football, Byron became the landlord of the New Inn public house on Elland Road, Churwell, Leeds, which had been managed by another Leeds United and Wales international legend John Charles years earlier.
In May 1978 Stevenson made his Wales debut at the Racecourse, Wrexham in the 1–0 win over Northern Ireland in the British Championship.