This twisted and skewered felt hat, from a private collection, has no label, but it seems like the logical (or illogical) result of hat patterns and illustrations from the late 1930s.
Here are three hats shown with couture collections in February, 1936.
Sketches of Paris Couture, Woman’s Home Companion, February 1936
The designers are, left to right, Mainbocher, Worth, and Molyneux. Fourteen months later, similar styles were available to home stitchers in a Butterick pattern.
Butterick Hat Pattern #7852: Four Hat Styles
Butterick # 7852, Butterick Fashion News, May 1938
“All in one pattern, you will find the four important hats of the season – the pill-box, the draped turban with height, the drapable, cone-shaped hat, and the brimmed bonnet. Designed for 21 ½ to 23 inches head. 25 Cents.”
Except for the pillbox hat (top left), three have pointed or flattened cone shapes, which had been appearing at least since 1936.
Ad, January 1936, and Pattern Illustration, December 1936, WHC
Here are several other cone hats, from 1937:
Two Views, Pattern Illustration, Woman’s Home Companion, October 1937
The version above is made from black Persian lamb. (Woman’s Home Companion, October of 1937)
This black felt cone hat in the illustration above is from a story in Woman’s Home Companion, May, 1937.
December 1937, Woman’s Home Companion
This green draped hat appeared in a pattern illustration, Woman’s Home Companion, December 1937. It looks as though it might be an asymmetrical bow, but it is very similar to the draped cone hat in pattern # 7852, seen from a different angle.
Finally, the draped and skewered cone hat illustrated on the left, below, from October, 1936, is only a little less extreme than the draped and skewered cone hat we started with:
The one on the right ties behind the head. The one on the left seems to depend on magic… or a thin elastic band.
witness2fashion 
Now that is a real hatpin! It is interesting how closely the home sewing patterns mirrored the French couture models.
I didn’t give credit to Elsa Schiaparelli for starting the fashion for pointy hats — and hats with a flattened cone shape — because I only saw sketches of her 1933-34 winter collection yesterday, while looking up something else. Her friends in the Dada movement helped to publicize them — so absurdity was part of “the point.” (My reference is Shocking: The Art and Fashion of Elsa Schiaparelli, by Dilys Blum.)
I’ve never really understood hats–especially wild shapes like the ones here. But I’m learning. How great that you tracked down patterns for the current Parisian designs. I wouldn’t have guessed that a magazine like Woman’s Home Companion would be so fashion forward.
Butterick had a permanent office in Paris, so they could “sketch” the latest fashions, and WHC sold Companion-Butterick patterns. (It’s amazing how quickly copied clothes made it from the runway to the department stores, back when runway styles were wearable garments, rather than outrageous advertising for perfumes!)