"Kept under one hat" is an expression used when referring to something that is kept secret or hidden. The antonyms for this phrase would be openness, transparency, and disclosure. If something is kept under one hat, it means that it is not being shared with others, whereas if something is open and transparent, it is being shared for all to see. Disclosure implies that information has been revealed that was previously kept hidden, and is the opposite of "kept under one hat." Additionally, honesty, candor, and frankness are all antonyms relating to the concept of being open and truthful, opposing the concealment suggested by "kept under one hat.