Term for "[Idiom], [Rhyme]"

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]"?

I think there is no universal name for these phrases but there is a name that I have found (which might be too general also):

• rhyming catch-phrase

Also we can call the latter part of the phrase a rhyme-tag:

a word or phrase used primarily to produce a rhyme. Rhyme-tags are used to comic effect in much light verse, as in W.S. Gilbert’s “The Modern Major-Gineral,” which reads in part

Furthermore, The Book "Slang: The People's Poetry By Michael Adams" mentions these phrases in a variety of ways, as in rhyming {something}. Also talks about the differences between rhyming slang and these phrases.

Additionally, the author thinks that they are a kind of verse and calls them rhyming couplets.

From Wikipedia, an internal rhyme or middle rhyme is a rhyme that occurs within a single line of a verse.

Although the article in Wikipedia is not brilliant, I think the connection with hip hop and swing era lyrics is important and missing in your question

I've found other links that give examples closer to the ones you've listed:

• britannica
• yourdictionary.com