I don"t believe she can apply double jeopardy. First trial got her convicted. She appealed got it overturned. Higher court reversed first appeal. That would fly here and should
N... No, it wouldn't. That's procedurally impossible here, mostly because that's not what actually happened, but also because our system is more of a pass/fail with verdicts.
There isn't really a perfect analogue here, but the gist is that Italians get two trials. The appellate court is actually able to return two basic types of acquittals, which break down further into five actual verdicts. The point is that the second trial returned a full-on 'Not Guilty' (as opposed to a 'Not Proven') because of insufficiency. And blisteringly so: "Material nonexistence" of evidence in the first trial, a motive that "while not corroborated by any evidence, is itself far from probable," and ends with "Having excluded the existence of any proof of the guilt of the current defendants..."
In the US, the overturning of a guilty verdict resulting in a retrial is considered a continuation of jeopardy provided that it's a procedural issue (and a few other miscellaneous things). A completed trial with a conviction that is overturned on grounds of sufficiency cannot be retried. See Burks v. US.
So it's that second trial that was overturned. The second verdict is only supposed to be overturned on some kind of issue with procedure or the application of the law. There is allegedly much fishiness in how this appeal went. It has nothing to do with the evidence of the original trial, but with the trial of Guede determining that there were additional perpetrators. Let me say that again: The opinion of the court in a separate trial of another person was accepted as evidence in an appeal that is not supposed to be about evidence. That a third court is able to render an opinion in a criminal trial at all, much less have it accepted as "fact" in another trial is a complete joke.
The second trial was re-tried, and it returned a conviction. That can be appealed just like the last second trial.
Also, they don't have juries. Not like we do, anyway. It's kind of an odd system.
So yeah, I wouldn't give the Manson, either.