I'm looking for nursery rhymes or traditional songs that have a character named Jack in them.* I'm trying to avoid the most obvious candidates (Jack Be Nimble, Jack and Jill, Jack Sprat, Little Jack Horner, The House that Jack Built). Others I've found in my searching and am still considering are Jack-a-Nory, Little Jack Jingle, Jack-a-Dandy, Jack Rowland, Jack Tar, and Black Jack Davy, but I'm looking for other options so I can choose the one that works best. I've googled for combinations of "jack" and "nursery rhyme" and "ballad" and "folk song", but a) the most obvious candidates always come up and swamp the less common results, and b) especially in the folk song searches, I get more results for singers named Jack.
*Not for Graveyard Book fic (though I kind of wish it were). The scenario is that a character (Jayne from Firefly, to be precise) is playing cards while under the inadvertant effect of a mind-altering substance, and can't remember the name of that "guy with the little hat" in his hand, but comes out with a short verse that contains the word 'jack' instead. I'm envisioning this rhyme as one he might have learned as a child and would remember under duress, so something fairly simple and with obvious rhymes would be good.
ETA: Thanks for all the suggestions! I ended up going with the 'futurized' 'House That Jack Built' idea, because I loved it :)