Hayes. This Is the Bear

This Is the Bear is a children's book by Sarah Hayes, with art by Helen Craig, published in 1986. It is available at the Internet Archive.

Here is a YouTube read-along:

This is a very sweet book! It's not exactly a "Jack" type of chain, but it is told in very simple rhyme, and does have a chain of sorts as the story goes from the bear who fell in a bin, a dog who pushed him in, a man who picked up the sack, and the driver who would not come back. 

That's how the bear ends up in the trash dump: poor little bear! The rest of the story tells how the boy finally found his bear at the dump; it's not much of a chain, but it is a very sweet story, all in rhyming couplets.

Here's how the story winds down at the end, and notice how there is commentary from the characters in the illustrations, in addition to the rhyming couplets:

The only reason I'm giving this just two stars is because it's not fully a chain tale, but it was so cute, I still wanted to include it here.