What's the Idiom or typical expression when a person "takes a joke further"

To "run with" something is (OED)

to advance or proceed with (an idea, undertaking, etc.).

To "run with a joke" would therefore be to play along with a joke and add new ideas to it. Here's an example of the usage, from the Screencrush website:

[T]he straightforward horror movie was erroneously placed in the LGBT section of their content library. Some clever Twitter user nabbed the screenshot and jokingly suggested that the ‘B’ in LGBT stands for Babadook, and as is the wont of Twitter users, they eagerly ran with the joke. Fan art, every permutation of the word (the Babadook likes to get all dressed up for the Pride parades with a fresh Baba-look, and when stunned, often feels Baba-shook), and dozens of other stupidly clever gags sprung up out of this weird obsession, recasting a monster as something kinder and warmer. Just goes to show: never underestimate the internet’s ability to extend a joke far past the point of reason until it grows into a bona fide cultural phenomenon.

Although it may not be as joke specific as you want, "jumping the shark" does get at pushing the comedy beyond its original effectiveness.

"Jumping the shark" is attempting to draw attention to or create publicity for something that is perceived as not warranting the attention, especially something that is believed to be past its peak in quality or relevance. The idiom "jumping the shark" is pejorative, most commonly used in reference to gimmicks for promoting entertainment outlets, such as a television series, that are declining in popularity.[1] https://en.wikipedia.org/wiki/Jumping_the_shark

You could use the idioms of "leapfrogging" or "piggybacking" to imply that the latter joke builds off the former humorous content. You could also refer to the escalated humor as being "stacked" or "stacked humor."