I think Double Jeopardy doesn't apply here.
Under the Fifth Amendment of the U.S. Constitution, prosecutors may not try a defendant for the same crime twice. For this reason, a person that is found not guilty by a jury cannot be then forced to face trial a second time.
The double jeopardy protection does not apply, however, to a case where the defendant has successfully appealed his conviction. In such cases, prosecutors do have the right to retry a defendant a second time, although they must do so in light of whatever issues the appeals court relied upon in overturning the conviction.
When Prosecutors Will Retry After an Appeal
Whether a defendant will face retrial after an appeal is obviously an important question to consider, as a second conviction may deny that person the benefit of their efforts to appeal. Furthermore, a second conviction can include a second criminal sentence which may be more or less harsh than the original sentence.
Ultimately, the decision of whether or not to retry after a conviction is completely within the discretion of the prosecutor’s office that had originally tried the case. In general, prosecutors bring cases on retrial when they think they have enough evidence to win, the potential for a sympathetic jury, and there is enough public sentiment in favor of retrial.
Very few prosecutors intentionally bring cases they think they will lose, and so the decision on whether to conduct a retrial will be based on a variety of factors, but one of the key factors will be in whether they can believe they can score another conviction in light of the appeals court decision. If the appeals court threw out a main piece of evidence used in prosecuting the case, such as a confession or the admission of contraband, the prosecutors may not have the ability to win without that evidence, and thus would decline to retry. Likewise, if the court ruled that the jury was given incorrect instructions on the law, and the correct reading of the law would never support a second conviction, this may be another reason not to retry.