The most important thing in asking a good follow-up question is to LISTEN to the response the interview subject gives to your prior question.
Many beginners make the mistake of preparing a list of questions, and then simply asking them one after the other, not really paying attention to how they’re answered. Preparing questions ahead of time is good, whether you actually write them down, or carry them in your head. The more time you have for preparation, the better. But your list should only be a flexible outline, not a “shooting script”.
If there’s a question you feel you absolutely MUST ask an interview subject, ask it first. That way you won’t be trying to remember to be sure to ask it, instead of actively listening to the responses your subject is giving.
Another beginner’s mistake is failure to ask the follow-up question “what did you mean by that?” If you don’t understand the answer, ask for clarification. If you’ve done your homework, and you get a response you don’t understand, how will your viewer or listener be able to understand? Experts in any field, whether it’s public administration or rocket science, will often use technical terms or jargon which their peers will readily understand, but the average person won’t. Don’t be afraid to ask for clarification on follow-up. You won’t look dumb. But you will if you write a story about something you don’t understand.