"Always look at the back view of your tentative hat. Sometimes, wide 'sustaining' bands will spoil the look of your hair; streamers can look ridiculous; and sometimes a dreamy sweep of straw will crumble into a linear apology in the back. Be stringent. Never buy a hat that is becoming from only one angle."
Just as you would check the back and side views if you were buying a dress, you should check those views when buying a hat.
The hat above is a white felt postilion by Dobbs, from January 1941.
What's a sustaining band? Many hats from the late 30's into the 40's were worn tipped forward over the brow and needed a band slipped over the back of the head to hold them in place. Often times, it was just a thin elastic cord, but I've also seen many with thicker bands like this.