This particular type of head wear from the middle ages is called a Chaperon. You can wear it a variety of different ways, but it is basically a pointed hood with a wide collar. The collar can have different embellishment (pointed, dogged, trefoil edges with embroidery etc.).

This lora was made using photos from re-enactors/living historians/re-faire folks.... so the subjects are usually outdoors and wearing wool or linen clothing. The chaperon can be worn with the hood down, but that might be another lora (or a version down the line) so right now its "chaperon up"

The long tail on some chaperons is called a liripipe. Some of the photos I used had long liripipes, in which case I tagged it as chaperon liripipe.

Also, the Chaperon can be rolled up and worn more like a hat... a few photos I used had that so I tagged it as a "chaperon rolled".

You should be able to put any color name you want with chaperon ie white chaperon
The overall trigger is chaperon, but here are some of the tags I used.

Chaperon
Chaperon up, wool chaperon, linen chaperon, chaperon liripipe,

I have another Chaperon lora for Illustrious... but this one does realism pretty well with SDXL

Have fun and good luck!