Blackmail, 1939, Edward G. Robinson. A fugitive from a chain gang becomes an oil-well firefighter and meets the man who framed him.
Blackwell's Island, 1939, John Garfield. A reporter sends a mobster to prison, then goes in after him posing as a convict.
Castle on the Hudson, 1940, John Garfield. Sing Sing's warden lets a convict visit his girlfriend on the honor system, but something goes wrong.
Each Dawn I Die, 1939, James Cagney. An innocent reporter is found guilty of murder and is sent to jail while his friends at the newspaper try to find out who framed him.
I Am a Fugitive from a Chain Gang, 1932, Paul Muni. An innocent man is sent to a prison farm, where he endures the dehumanizing effects of a cruel criminal justice system.
Invisible Stripes, 1939, George Raft. An ex-convict joins bank robbers t finance his younger brother's honest business.
These are what I can think of at the moment. There were a lot of them made in the 1930's starring Robinson, Raft, Bogart, Garfield, Muni, etc.