"There was an old man and he wasn't very rich, And when he died, he didn't leave much But a great big hat with a great big rim All bound 'round with a woolen string. A woolen string (x2), All bound round... A great big hat with a... All bound round...."
Linscott believes the words to this to be related to "All Around My Hat." I don't see the resemblance; it made me think of "The Miller's Three Sons." The tune is said to be related to the Irish air "Old Rose Tree." - RBW