Most likely, a party injured by the new law would file a suit challenging its constitutionality. The suit would proceed through the court system, eventually reaching the Supreme Court. SCOTUS would then be free to rule on the issue at hand.
Assume your premise that 100% of the House and Senate vote for a bill, and the President signs it. What makes you think that prevents a Supreme Court review? It does no such thing.
Any person, any group, that think the law is unconstitutional can get a lawyer, go to court, and challenge. If they have enough money, and patience, they can take the matter to the Supreme Court. And if the Court agrees with them. the law is then dead.
You're talking to someone here who has, along the way, sued Ford, Carter, ABC, CBS, NBC, the FEC, and the League of Women Voters, among others, God Himself could not stop a court review of any law, unless he followed Shakespeare's advice, "First, kill all the lawyers."
Is that clear enough?
Billybob