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If you want to get new evidence into a case, you need to get the case back to a trial, or pre-trial phase.
You can use new evidence in support of your motion (for example, if you were to file a "Motion for Reconsideration" based on newly discovered evidence, you would have to submit the new evidence in support of the motion, the court would not consider the evidence for the merits of the evidence itself, but would consider it in support of the motion).
Similarly, in an appeal, you can ask the appellate court to consider evidence that the trial court had rejected as being important to the underlying proceeding. While the appellate court does not evaluate the sufficiency of the underlying evidence, they still need to know what you are talking about, so they will look at it to support the appeal. If you prevail on appeal, they will send the matter back down to the trial court with instructions to reconsider the matter including the new evidence.