Haunted Harbor (1944) - Kane Richmond
This is a 15 Chapter Republic movie serial, based on the novel by Ewart Adamson. The serial was re-released in 1951, under the new title Pirates' Harbor.
On the island of Amoa, Jim Marsden (Kane Richmond) is falsely accused of a murder and nearly executed but he escapes with the help of some friends. Sailing to another island, which, oddly, resembles California, he encounters Patricia Harding (Kay Aldridge) and her father and saves them from drowning. The actual murderer, Carter (Roy Barcroft), winds up on the same island as Marsden and has taken the identity of Kane.
NOTE: In the link to access the serial Chapters, the words Part are the same as Chapter.