Umbrella.
And a better hat ...
Since it is a 50 or so year old hat - could it just be water-soluble microsized dust & dirt getting washed out? My thinking is that a gradual build-up of dirt would be less likely to be noticed on brown since it's the same color. On a relatively new hat, I'd have to think it's just excess dye. But I know I've hand-washed a few things that looked pretty clean and was surprised at how dirty the washwater was.
My question is had the hat ever been cleaned?