Many characters don't know the Old Mayor of Little Marrow and those who do know him, like the Lighthouse Keeper, do not want talk about him.

The two characters that do talk about the old mayor are the Grieving Father and the Mayor. According to the two, the old mayor went mad and spent his final days pacing the shoreline and screaming "Throw it back!" while throwing people's belongings into the sea. After a few days of that, he disappeared from the Marrows and his tale is used to scare children.

Plot or ending details follow.
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What little the Lighthouse Keeper says about him makes it clear the old mayor had a boat of his own, specifically a rowboat. So it's comes to no surprise to learn that unlike other characters, he doesn't have a fixed location. He moves between his campsite that can be found near the four makeshift docks. Unfamiliar symbols are often found in his campsite, these symbols are most visible in his Stellar Basin camp and Twisted Strand camp.

Meeting the old mayor is needed to unlock the alternate ending.

When you find the old mayor, he will be hunched by a campfire muttering to himself. He will ask the Fisherman if he had thrown the book back, seeing the fisherman's confusion. He will then talk about a man and his wife, who dredged up the book from the sea and along with it was something dreadful cloaked in fog. This unknown thing sunk the ship they were on. They washed up on shore but that man was still clutching the book. They pleaded with him to throw the book back as the fog rolled on to shore.

After that the old mayor will tell the fisherman to talk to the Lighthouse Keeper. Saying she saw that ruinous day and knows where the book must be returned to.