You usually do not need to restart the entire answer. Add the missing context briefly and continue.

Fit answers work best when the interviewer can quickly understand the situation, your role, the tension, and what you personally did.

If you notice the gap, a short correction such as "One piece of context I should add" is cleaner than abandoning the story and starting again.

Practice setting the scene in one or two sentences so the action and impact remain the focus. case interview guide.

I have done this in mocks. A quick "one thing I should clarify" sounds way less awkward than restarting.