Doubling down to save face is how small mistakes become famous ones.

A small error is forgettable until you defend it. Defending it adds a second decision, this time a visible one, and now there is something to argue about with an audience. Each round makes a retreat more expensive, until the thing everyone remembers is not the mistake at all but how long you fought for it.

When you notice you are wrong, say it early and plainly, while it is still small enough that nobody cares. One sentence, no cushioning, no explanation of why it was reasonable. The speed of the admission is what keeps the story from forming.