Is it his hair?
What if I told you that Captain Rufus Tucker Compson has never been seen because he died in the war, and Bud keeps his father's medal in his hat to remember him by.
But that would be a lie, Dave.
Wouldn't it be interesting, though?
Maybe take it a step further, and his hat actually belonged to Rufus. He wears it in his honor like Boba Fett wearing Jango Fett's helmet after he was killed.