I don't know why I didn't notice the waves of coverage before, but it helps to be aware to have appropriate talking points for each wave.
1. Factual articles: motivation for paper, what it adds, main findings, conclusion, take-away point. Mostly in print media.
2. Reaction articles and 5-10 minutes radio interviews. Presumably people have already seen the factual articles in whole or part, so findings are presented briefly by someone else. They want to know findings in 1 sentence, take-away point in 1-2 sentences, why for each finding. Then they ask a bunch of questions well beyond the scope of your paper, and if you answered the questions as asked you would be speculating, so you need things to say that you are qualified to say related to their questions.
My favorite question from a radio interview, "Do couples who have sex have better relationships than couples who don't have sex?" I still don't know what I could have said.
Special interest press will pick out a tangential issue and expand on it so it seems like your study was about the background issue.
3. Call-in shows are like radio interviewers with more personal examples. The personal examples may line up exactly with your take-home points, which is extremely gratifying, or they may be interesting but something you only have anecdotal experience with.
4. Reactions to reactions: Party X raises objection Y. Talking point: One 20 word quote if it's a reasonable objection.
The hardest part, I find, is expanding the limits of what one can say. Medical journal papers are so extremely circumscribed, while press coverage asks about all kinds of issues that one could never justifiably publish in a medical journal without a whole separate study that one didn't do, so you just have to quote literature over and over.