Instead of electing the POTUS, who then appoints a cabinet, we (optionally) eliminate the office of the President and instead elect the cabinet level leaders directly. They would still be part of the executive branch of government. We would promote the involvement of minor parties and essentially eliminate the Democratic and Republican parties, which we’ve seen are irretrevably corrupted by money.
Here’s what the Cabinet makeup could look like. These might even be permanent assignments with different people elected to the offices after expiration of terms. Here there has been a quick attempt to balance competing offices with different parties.
I don’t know who would represent Green, Socialist and Libertarians. Maybe that would be left to a convention of their splintered parties. The US doesn’t have Pirate and Science parties, which would be modeled after the respective parties in Iceland and Australia.