If he was a double murderer, plotting to and successfully escaping isn't a redemption, it's just a murderer getting away with it.
The way I remember thinking about it was that he was jailed for revenge murder, then spent his life in jail doing his best to atone by being helpful (building a library, teaching, helping with taxes, etc.). When the prison system refuses to set him free despite him proving through his actions in prison that he's not a threat to society anymore (I hallucinated this part -- this happened to Red, not Andy), he escapes, and his freedom is his redemption.
I'm not a native English speaker, and I think I may have conflated redemption and atonement. Looking at some definitions, it looks like you can receive redemption without atonement -- it doesn't necessarily have to come from within.
A more recent prison movie which made me feel similarly to Cool Hand Luke and Shawshank Redemption while watching it is "I Love You Phillip Morris" (starring Jim Carrey and Ewan McGregor).