Essentially, you're asking, "How can I become a 'punnier' person?" (Or, perhaps more specifically, "What techniques might help me conjure witty puns, rhymes, and play-on-words wordplay more easily?")
This is a topic I've mused on for some time. I haven't done any formal research on the subject, but I've observed that some people can pen their parody, poetry, and puns almost effortlessly – their Groucho Marxist quips pop into their minds just about automatically, as if they are hardwired to think of "oxyneuronic" thoughts – while others might flail to formulate a similarly witty idea, with their results coming across as flat and forced.
Can someone become self-trained to become more adroit with such wit? My first suggestion would be to immerse yourself in the world of borrowed- and double-meaning. If you want to become fluent in French, move to France; if you want to become a more creative tagline punster, surround yourself with the cream of the word-croppers. I'd recommend that you start by working on first-rate crossword puzzles. Puzzle makers do a great job of using secondary meanings of words to create clues that seem almost misleading at first. Pay particular attention to clues with a question-mark at the end, as these are used to indicate a stretched pun. As one website says:
A question mark at the end of a clue usually signals that the clue/answer combination involves some sort of pun, e.g. "Grateful?" = ASHES
Will that work? In other words, will such exercises help train your brain to be more creative? I don't know for sure, but at least it's something practical you can try. That beats handwringing, at least in my book.
I'd also pay attention to pop culture. I think the most effective puns in advertising are references to widely-recognized idioms and expressions. For example, "Take me to an Economist reader" is a pun off of the more well-known "Take me to your leader," which comes from cheesy sci-fi movies. But if you're not in tune with what's trending, you'll be unaware of a trove potential catchphrases to play off of.