Does it have to be a die cut, or could you punchart or make your own card?
Here's a pattern to go by: Make Your Own Nurse Cap - Free Pattern and Directions to Create a Nurse's Cap
If you're trying to make a card you could measure out how wide and tall you want. Score in the middle and fold. You'd want the fold to be on the left side. Allowing for the flared square part of the hat (where your head would go), measure down and pencil a rectangle across your card. Open the card, keeping in mind that the right side will be the front. Either punch out or die cut part of the top,using the widest
part of a large oval punch/die, or if they are too small you could scan and enlarge a punchout of the oval as your template. Do the same for the bottom, only use a circle
punch/die. Remember: the idea is to only take parts of the card out, not the entire punch/die cut shape.
Now for the back (left) side: measure how much of a flared square looks good proportionally with the rest of the card. Depending on the overall size of the hat/card, add 1/8 - 1/4 (or more) inches to the lower portion of the sides of the square (translation: the top part will be a square and the bottom will be slightly wider on each side).
Using the cut off portion of card from your front, place the oval parts just below the square bottom. Trace the oval from the sides of the card to the square (on either side... translation: you want a partial oval - such as you did on the front of the card - but not going all the way across, as you want the square to be in the center).
Do the same on the bottom of the card as you did on the front: use your circle punch/die to remove a portion of the card.
Look at the picture from the link provided... it'll make more sense. You want a boat shaped card with the square portion on the back part of the card sticking up.
If you had intended to have a hat as a 3D or to put onto a card, the punchart would be a lot easier to do.
I hope that you can make sense of this post or at least give you an idea with which to work. Good luck!
Here's another link, using svgcuts file (can be downloaded): Paper Loving Critter: Nurse Cap Shape Card
p.s. - Here's another link, using svgcuts file (can be downloaded): Paper Loving Critter: Nurse Cap Shape Card