keep under hat
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Related to keep under hat: keep under one's hat
keep (something) under (one's) hat
To not reveal or discuss something publicly. Keep this under your hat for now, but Greg is definitely the one getting the promotion.
keep something under one's hat
Fig. to keep something a secret; to keep something in one's mind (only). (If the secret stays under your hat, it stays in your mind. Note the use of but in the examples.) Keep this under your hat, but I'm getting married. I'm getting married, but keep it under your hat.