Okay, Zimmerman is a bad example because see Mr. Stark, he didn't deny doing the killing on the stand, merely argued it was legal. But I do presume that double jeopardy extends to forbidding perjury charges on general denials somebody did a crime, or even false alibis, if the State later gets more evidence the person really committed the crime and encourages the jury to use the perjury charge as a stand-in for re-charging them with the same crime.
That said, what about perjury charges for something different than general denials of responsibility? e.g. Alice says on the stand that sure, she was at Bob's house, but she saw *Carol* shoot Bob, so she's innocent. In a display of prosecutorial incompetence, the prosecutor doesn't closely probe into this accusation, and Alice is acquitted. Carol later shows up, angry at being defamed, and has an iron-clad alibi of being in another state with lots of witnesses, so Alice was clearly lying. Can Alice be prosecuted for perjury? (I'd assume no, but.)