After you start and exit the cave, head straight and a bit to the left, eventually you will hit a wall where there will be a gap. Jump through the gap and you will see a river in front of you, follow the river and you will find a Public Lake House. Head into the house through the stairs and on the opposite side of the stairs there will be a dock like area. Follow the dock to it's end and you will find the hat that you can equip.