What is the term for finding words within words for comedic effect?

Question

I am trying to find a word that describes finding a word within another word for comedic effect i.e. functioning as a pun or word play.

The example that sparked this question was when I came across a product name written on the box for hair regrowth in men. The text was "Hair A-Gain".

In this example, the noun/verb gain can be found within the adverb again; it is clever because it refers to two separate but related ideas → having hair once more as well as the act of "gaining" more hair.

What is the correct term for this phenomenon of finding words within words for comedic effect?

EDIT: Follow up question → what is simply the correct term describing the phenomenon of finding words within words?

Answer 1

It is a "play-on-words" or "word play". It is a formal form of "wit", though here no pun intended. Another form of it might be a "pun", in this case, pun intended.

https://idioms.thefreedictionary.com/play+on+words

https://en.wikipedia.org/wiki/Word_play

https://www.thefreedictionary.com/pun

https://www.urbandictionary.com/define.php?term=no%20pun%20intended