a crush-hat [coined English-French, not real French]
Chambers's Twentieth Century Dictionary (1908)
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a hat so made that it can be compressed and carried under the arm without injury. Such hats were particularly worn on dress occasions by gentlemen in the 18th century. A chapeau bras is now worn in the United States army by general and staff officers.
"I am ready to go," she said, taking his arm. At the door of the box she turned and looked straight at Chauvelin, who, with his chapeau-bras under his arm, and a curious smile round his thin lips, was preparing to follow the strangely ill-assorted couple.
Baroness Emmuska Orczy. The Scarlet Pimpernel (1905)
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Under one arm he carried a huge chapeau-de-bras, and under the other a fiddle nearly five times as big as himself.
Edgar Allan Poe. The Works of Edgar Allan Poe — Volume 4: The Devil in the Belfry (1839)