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?