No, it's not common to reopen court cases if the jury finds a person innocent. You can certainly reopen court cases where someone was found guilty, but that's not double jeopardy.
Innocent = "We, the jury, are certain the defendant didn't do it."
Not guilty = "The prosecution did not convince us, the jury, that the defendant did it."
And because our criminal justice system is not set up to establish innocence, only to check if there's enough evidence to prove guilt, we don't use the term "innocent" for decisions.