A head-covering apparatus is disclosed, which extends over the top of a human head and down its sides and toward the back of the neck. The apparatus has a forward extending portion with two pocket-style openings, the first pocket for receiving a flexible material that makes up the head-covering portion of the apparatus, and the second pocket for receiving the visor or bill of a baseball-style hat. The apparatus can be affixed to the visor/bill of a baseball-style hat and used in two different configurations: in the first configuration, the invention is “unfurled” so that its flexible material covers the human head and portions of the baseball hat; in the second configuration, the invention is compacted such that its flexible material is rolled or folded into a small compacted size, and then placed into the second pocket of the forward extending portion.