Phantom Lady (1942)

From Wikipedia: "A man is first accused, and then convicted, of murdering his wife. As his execution date approaches, his friends and a sympathetic detective frantically search for his alibi, a woman with whom he'd gone to a Broadway show the night of the murder. None of the people who saw them together recall the woman.She is a phantom in a city of millions who cannot be found and does not come forward. He was found guilty and sentenced to death and now sits in the death house at the prison. His only hope is now the small group of friends turning the city upside down trying to find this “Phantom Lady.” Each chapter of the book details the search efforts made to free him. Each chapter details how each effort fails. Each chapter describes a day closer to his execution."