During the later middle ages young women wore cone shaped head wear called "Hennins". These hennins were either single or double cones worn on the head and sometimes draped in a veil. This often hall hat was held on the wearer's head by a small metal strap or tab that rested on the forehead to counteract the weight of the hat.
So, some of those medieval movies throughout history with their "princess hats" are somewhat correct! Maid Marion in Disney's 1973 animated Robin Hood wore a butterfly or double cone hennin. Her character designer, Ken Anderson, cheekily used her ears to create the shape.
Your dress up "princess hat" is actually older than the medieval times as mentioned in a Smithsonian Magazine article. It is believed, at least the single cone hennin was designed with heavy influences from the Mongol married women's head coverings known as Boqta. These headdresses sat more vertically and according to some historical accounts made it very difficult for Mongol women to enter and exit their tents! Most were made out of wool or flannel, but some of the richer women had them made with red silk.
Why did ancient and not as ancient women want such tall head pieces. I can only imagine the headaches that came along with them.
But, even more recent women have worn headpieces that added to their verticality. Fontages in the 1700s were so tall that some women DIED after they caught fire from the candelabras and chandeliers overhead. I guess what they say in Texas is right, "The taller the historical headpiece, the closer to the fashion god" ...or something like that.