I often find that some of the best story tidbits come out after the interview has officially ended. When the last prepared question has been asked and answered, interviewees tend to relax into comfortable conversation. They may elaborate with more emotion or candor on something they've already discussed with you. Quotes may become more meaningful and opinions more emphatic or emotional.
Sometimes, new story ideas on completely different topics may arise, or a new spin on the story you're working on will come to light.
So, even after the interview officially winds down, leave the voice recorder running. You'll be glad you did, and so will your readers.