What happens to Odysseus when they go hunting?
When Odysseus goes hunting in Book 10, he finds a great stag. He uses the meat to revive the energy and spirits of his men. After that is when he and his men encounter Circe and the men are transformed.
In Book 19, an old hunting scar of Odysseus' is revealed (he'd been gored by a boar), and it is used to identify him.