Sticking to (and building on) your story works if you do it over time.

If it happens, you'll stick to your story, no matter how awkward the situation becomes.

She paused, suddenly wondering whether to stick to her prepared story, or to change it.

"And you stick to your story that you didn't hear the shot?"

He stuck to his story of a Secret Service job.

They stuck to their story and it sounded straight to me.

I stuck to my story, and by the end of the day, I thought they might actually have believed me.

Or Clinton might stick to his story because it's true.

She is bound to stick to her story, or else confess everything.

But they're sticking to their story, and I don't blame them.