Public service announcement -- with the exception of rare cases, (look up original jurisdiction if you want to) the US SC does not entertain new evidence. It looks at the lower court ruling and listens to oral arguments and briefs based on the lower court rulings.

No one, no matter what, has to produce anything evidentiary to the SC. The SC could remand the case to the lower court and require more evidence be reviewed because their ruling couldn't have come to the same conclusion based on the available evidence, but the SC doesn't look for or review additional evidence on a Writ.