None. The US Supreme Court is the highest appellate court in the United States, and the ultimate arbiter of any case it hears. Matters decided by the Court are considered res judicata, meaning they have received final judgment and are no longer eligible for appeal.
The companion doctrine of collateral estoppel prevents either party from bringing a new suit related to any matter in the case decided. Neither party can seek to retry the case on different grounds.
The only exception is when the Supreme Court issues an opinion and chooses to remand (return) the case to a lower court for further action. This may occur if the Court determines certain evidence should have been admissible at trial, or if a judge made a procedural error that results in the case needing to be retried. Cases are remanded at the sole discretion of the Court.