Longman Dictionary of Contemporary English
Wikipedia
A hat box (also commonly hatbox and sometimes hat bucket or hat tin) is a container for storing and transporting headgear, protecting it from damage and dust. A more generic term for a box used to carry garments, including headgear, is a bandbox. Typically, a hat box is deep and round in shape, although it may also be boxlike and used as an item of luggage for transporting a variety of hats.
Hat boxes may be made of a range of materials, including cardboard, leather or metal. They may include straps or a carrying handle for transportation. More luxurious models may be padded and lined in materials such as silk in order to protect the headgear.
Usage examples of "hat box".
We don't search every trunk, cooler, and hat box coming across the Friendship Bridge, looking for Canadian terrorists with suitcase nukes.
For a few seconds they jiggled around like a pair of hippos in a hat box, until Troy said `We could all walk,' and they descended the steps to emerge below street level in a converted wine cellar.