Appellate courts are not finders of fact. Appeals courts are supposed to resolve questions of law. The job of a court of appeals isn’t supposed to be to second-guess whether the witnesses are telling the truth. Sometimes, as you mention, an appellate judge does point out that the factual record they’re presented with does not make sense, but, in theory, that isn’t what they’re reviewing.
The facts of the case are relevant to that one case. What the Supreme Court actually decided was not whether that one person had prayed too loudly or not, but that “The Free Exercise and Free Speech Clauses of the First Amendment protect an individual engaging in a personal religious observance from government reprisal.” That holding doesn’t depend in any way on the actual volume of that one plaintiff’s prayers. (All right, sometimes it can, because someone says, “If even that was legal, this has to be,” but if someone were worried about that, she would have been motivated to emphasize how very very exceptionally quiet this one prayer that’s allowable really was.) Similarly, the landmark case of Lawrence v. Texas was widely misreported in 1993 as beginning with a SWATTing of two gay men by a homophobic neighbor, when in fact it was a jealous boyfriend (although I see the opinion itself stuck to the passive voice and thus did not end up repeating it). This didn’t matter to the conclusion that sodomy laws against consenting adults in private violate the Constitution.
This is where the maxim, “Tough cases make bad law,” come from. Lawyers love to hand-pick the most sympathetic petitioners or the most unsympathetic defendants imaginable, so that it will feel like a miscarriage of justice if their side doesn’t win this specific case. But courts of appeals are actually establishing rules for how other cases must be decided in the future. The reasoning they come up with to justify one verdict in an unusual case often turns out to be a bad precedent for other cases.
The Supreme Court isn’t a trial court where people present new evidence. More precisely, the side that lost makes a list of all the errors they claim the judge made that they wnt to be overruled, and the higher court decides whether it will hear the arguments for and against. This is why the Supreme Court only has thirty-minute question periods: both parties to the case already submitted their written briefs where they lay out the case, and now the justices will ask any further questions that they feel were not sufficiently addressed. At this argument, the lawyers never say, “I have new evidence to introduce!” or “I want to call a witness!” It’s not a trial where they do that.
The higher court might rule that the trial judge erred by not allowing one side to present evidence, or it might rule that the trial judge erred by allowing a jury to hear something it should not have. If that happens, it orders there to be a new trial, with a new jury, that gets to hear the new evidence and decide based on that.
There’s no other court to appeal to from a decision by the Supreme Court. (Hence the famous quip, “We’re not last because we’re right; we’re right because we’re last.”) @DavePhD informed me that it can be petitioned to rehear a case it just decided, with a 25-day time limit.
However, it has happened that a case raised an issue of law that made the Supreme Court order a retrial, and in the new trial, a different question of law came up that also was appealed to the Supreme Court, so it heard the same case a second time. Powell v. Alabama, the trial of the Scottsboro Nine, was one such case, establishing two different precedents (on the right to effective assistance of counsel and the unconstitutionality of calling in only White people for jury duty). With the judge in one of these retrials throwing out the jury’s guilty verdict and declaring a mistrial because he was so sure their conviction had been wrongful, some of the nine defendants were tried four times. (On the same evidence, four of them ended up being acquitted, four were convicted, and one died in police custody.)
But, once a trial is over and its appeals are exhausted, the case is done. It cannot be tried again. This is called res judicata, a thing that has been judged. Norma McCorvey, better known as the pseudonymous “Jane Roe” in Roe v. Wade, even sued years afterward to have her own case overturned. (Too soon?) Notably, the holdings of that case were later overruled—in a separate case that raised the same issues and led the justices on today’s Supreme Court to reconsider them. But the original case was not, and could not be, brought a second time to undo it, and someone tried. Similarly, in another example I gave above, the court in Lawrence v. Texas reversed the conclusion it had reached in Bowers v. Hardwick, but without changing the outcome of the previous case (where the prosecutor ended up dropping the sodomy charge anyway).
