Redford was found dead in a wooded area in Irvine, Ayrshire, on Friday morning. It is understood there were no suspicious circumstances.
Redford moved from Dundee to Rangers for a then Scottish record fee of £210,000 in 1980, where he later teamed up with McCoist.
He made more than 200 appearances for Rangers and won the Scottish Cup and two League Cups, the first when he and the late Davie Cooper scored against Dundee United in the final.
McCoist told www.rangers.co.uk: "We are deeply shocked to hear the very sad news that Ian has passed away.
"I had the honour of playing with him early in my Rangers career and he was a terrific lad and an extremely talented footballer who will be sadly missed.
"Everyone at Rangers sends their sincere condolences to Ian's family and our thoughts are with them at this very difficult time."
Redford, who hid the fact he was deaf in one ear throughout his career, moved from Ibrox to Dundee United, where he helped the club to the UEFA Cup final with home and away wins over Barcelona along the way.
A statement from the Tannadice club read: "Everybody at Dundee United is devastated to hear of the passing of ex-United player Ian Redford.
"The memory of his performances for United at a time when the club competed in the higher echelons of European football will forever be part of our history and his winning goal against Borussia Monchengladbach in the UEFA Cup semi final in 1987 will always be looked back on with particular fondness by Arabs everywhere.
"Our thoughts and prayers are with Ian's family at this very sad time."
Dundee and Raith Rovers, where Perthshire-born Redford won another League Cup winners' medal in 1994, also paid tribute to their former player.
Dundee said: "Ian was a silky and attacking player who was not quite the stereotypical footballer - he'll be missed. Our thoughts are with his family and friends at this sad time."
The Kirkcaldy club said: "Although he joined Rovers in the twilight of his career, and played only 15 matches, his experience and influence on the talented youngsters around him was immense.
"The League Cup winner's medal was the swansong to a glittering career."