The exact same thing happened to me during a product interview. I bumbled through product design with the PM and could tell things weren't going well based on the questions she was throwing back at me, the awkward silences, and the increasing number of logical traps I found myself in as the interview progressed.

After the interview, I started beating myself up over the dumb things I said. But after a few hours, I opened up Photoshop, mocked up a better solution and sent it over to the recruiter (who I then asked to forward my solution to my interviewer).

I progressed to the next round.

I honestly don't know if my revised solution helped or not, but I'm sure it didn't hurt.