To doff a hat or other piece of clothing is to take it off.

// They doffed their coats when they came inside the house.

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“On the ferry from Oakland to San Francisco [Oscar] Wilde was introduced to a group of reporters who courteously doffed their hats. Wilde failed to return the gesture, much to the annoyance of one interviewer who used it as a pretext for blasting Wilde in his article.” — Rob Marland, LitHub.com, 11 Mar. 2024

Time was, people talked about doffing and donning articles of clothing with about the same frequency. But in the mid-19th century the verb don became significantly more popular and left doff to flounder a bit in linguistic semi-obscurity. Doff and don have been a pair from the start: both date to the 14th century, with doff arising as a Middle English contraction of the phrase “to do off” and don as a contraction of “to do on.” Shakespeare was among the first, as far as we know, to use the word as it’s defined in the more general sense of “to rid oneself of” or “put aside.” He has Juliet give voice to this sense when she says, “What’s in a name? That which we call a rose / By any other name would smell as sweet. / … Romeo, doff thy name; / And for that name, which is no part of thee, / Take all myself.”