Browsing the web, I came across this image of the cast of the television show Community using a type of structure familiar to me, and I wondered if there's a name for it.
The most familiar instance is likely "See you later, alligator." And because it brings to memory the similar "In a while, crocodile," I found a phrase that loosely encapsulated these with the Wikipage for Parting Phrases.
Then I remembered other examples from the musical number Telephone Hour in the 1963 film Bye Bye Birdie, in which teenyboppers greet each other on the phone with phrases like
What's the story, morning glory? What's [the] tale, nightingale? ... What's the word, humming bird?
And I've found several pages of lesser-known but valid examples of what I wish to describe, some examples being "So long, King [Donkey?] Kong" and "Mañana, iguana." So I was tempted to label this question "Term for '[Salutation], [Rhyming Organism]'."
Now I wanted the term also to include the quotes from Community, most of which (an example being "You're on your own, Al Capone") are preceded by a phrase that is not a salutation. But growing up we had a book called "See You Later, Alligator"; it contained the alligator and crocodile phrases, but it also had ones that were something like "See you on the veranda, panda" and "See you in pajamas, llamas." Because these opening phrases are not idioms or figures-of-speech, my ideal term does not include them.
And even organism may not quite capture the terminal phrase, as shown in Jeebs' example using a title in 2002's Men in Black II: "Let's make it happen, cap'n." So I have generalized it to rhyme.
The idiom restriction aside, I suspect there may be a term like iambic pentameter or Oom-Pah-Pah to describe the rhythm these phrases make when spoken. I am open to suggestions like these but my preference lies with terms more like antimetabole, a word I recently learned from ELU which focuses more on the structure or composition of the phrase than on its sound.
Is there a word or phrase that precisely describes expressions of the form "[Idiom], [Rhyming Phrase]"?