Lauren,
I agree with your dating of the early 1960s, and could even be very late 1950s. I might call that style a toque, which is a catch-all term that defines a variety of shapes an styles which are similar to your hat. It was likely intended to sit more on the back of the head, off the forehead completely and showing some small bit of hair in front, covering the "up do" or teased hairstyle towards the back. However, I prefer it the way you show it, which looks so much more stylish. I often show a hat worn in way that is more modern and stylish, and note in the description that it was intended to be worn differently in the time in which it was made. That way the buyer can decide and it is not misleading.