There are a number of ways to approach this. Perhaps the original way was described in Waite's Pictorial Key to the Tarot, but he limited it to situations where he got a court card as the outcome. Since he used Significator cards, he suggested using the outcome card as the Significator in an entirely new reading to show how the original matter would be influenced by that other person. The "short-cut" method of choice these days seems to be to pull clarifying cards for any perplexing card in a spread. I've been opposed to clarifiers in most cases because I think they muddy the original answer more often than they illuminate it, and they discourage (or at least complicate) deep thought on exactly what the first card was trying to say. A third way that splits it down the middle is the one I proposed in my "Rest of the Story" spread, which branches the initial outcome into three "what if" paths: "What will happen if I actively support or oppose the outcome, or simply do nothing?"
It can also depend on your standard practice for setting the effective duration of a reading. For example, I have a rule of thumb that views the Celtic Cross as "good" for a period of a few months, after which elements of chaos (unpredictability) can start to erode or dilute the precision of the original reading. Of course, this can happen with a reading of any duration in highly fluid or volatile circumstances, and if you're chasing a rapidly moving target further readings may help to nail it down. However, it should be driven by situational factors, not merely personal anxiety over getting unfavorable cards.
In short, though, as has been said, as long as you deconstruct your original question into different "angles of attack" that explore a range of variables in the matter, I don't see any real problem with it. I might suggest that you try using different decks to ask follow-up questions, if only to get a fresh perspective from the cards themselves.