Contrary to what most people think, the loop isn't attached to the hennin or pined in the hair, it's simply attached to a headband that's worn under the coif or directly under the hennin..

What's the purpuse of this thing you ask ?
Wow, these people were geniuses. I used to only wear a coif underneat my heavy hood chaperons and once the darn thing was starting to slip back, I had to cramp my finger underneat to reach the coif and pull it back in place. This process would often get a whole bunch of frizzy hair out and I'd often take the whole thing off all together and pin it in place all over again. Than I decided to make myself a silly little headband with a loop just for appearences, well it turns out that the little thing is a real miracle. When made of wool or velvet and tightly adjusted to the head, it grips to the hair like velcro so you're sure that it doesn't move all day. Then you either pin your coif to it or your hennin. When the weight of the hennin or chaperon manages to pull it backwards, you simply grab the loop and pull it back in place, it works so great that I could hardly believe it myself. No nore digging underneat to try and reach the coif !!!

On the first image you can see the loop-band worn with a truncated hennin, beggins or by itself. The second image confirms that a band is worn underneat the hennin. The third image indicated that the band and a small truncated form is worn underneat the beggin type of headdress (in the corner you can see the same lady with the headdress on from the previous page in the same manuscript).
Home page
References for Burgundian costuming
The turn-over brim
Here's my theory about the black turnover on the burgundian hennins...
If you look at the frist image below, you'll notice that the hood (coif, beggin or chaperon), is exactly the same shape as the turnover (brim) on the hennin beside. For that reason, I believe that in many cases, the noble ladies simply wore a conical hat over their regular hood or beggin. Later, the hennin became taller and they began styling and pining the flaps to the hat giving it various shapes and styles. The image represents the 3 classes of society. The second image is another reason why I believe this is the case. Both images represent the same woman from the same manuscript. In the first scene, the man orders his wife to take care of the kitchen as it's a woman's duty to take care of the house. In the second one, she's sweeping the floor. She's dressed exacly the same way, though the hennin has disapeared...
"Augustine, La Cité de Dieu" illuminated by maître François
Barthelemy l'Anglais "Livre des propriétés des choses" (property of things)
Bliss' No Right Clicking
How the loop is attached
Now speaking from experience
detail of the right wing of the "Triptych of the family Moreel" by Hans Memling
Detail of a portrait of Isabelle de Bourbon
Image from "Romance of the Rose" version from the British library
The brim and the loop
From the manuscript "Renaud de Montauban"
*Images 1 and 3 are from "Augustine, La Cité de Dieu" illuminated by maître François
*Image 2 is from Pierre le Baud's "compilation des chroniques & histoires des bretons 1480
*Image 4 is from a painting by Dierick Bouts called "Justice of the emperor Otton III" 1460
*Image 5 is from David Aubert's "Renaud de Montauban" (the banquet sceen) at the Arsenal library
Here are a few more exemples where you can clearly see a separate layer underneat the hennin.

With all the evidence that points to a headband, do I believe that it could be anything else than a band, like a cap for instance? Nop, not even remotly, not a chance, not in a million years...
So OK, this isn't a hennin but it still clearly shows the distinctive headband underneat the reticulated headdress which is still from 1478 by the way...
Hugo van der Goes, portrait of Margaret of Danemark 1478