"Double Jeopardy" only means being tried for the same, exact crime, after a prior completed trial.
They charge you, try you, you get a "not guilty" (as in, the jury finds that state failed to prove the case beyond a reasonable doubt), then they find more evidence- they cannot try you again for the same crime.
More to it than that, but that's the essence. I'll defer to more expert criminal law practitioners (Adam? David?) for a better description.