I’ve been working on a replica of the Monmouth cap for a while. The only extant original is in the Nelson Museum.*
My first attempt was based on several published patterns that are frequently referenced. You can view them here and here.

Using our new, bulky, pure Black Welsh yarn, I did the provisional cast-on. I knit for a bit, then dit a purl row, then knit again, then joined the sections at the provisional cast to make the double brim.
My second attempt was done more in line with my usual knitting style. I started without any pattern to speak of. This version went from the top down, as to me, the brim ledge looks like a cast-off, not a cast-on or join. I did a few areas of three-stitch i-cord to make the button, then did a series of increases and plain knitting until it was the right size.

For the increases, I did a knit in the front, then knit in the back of the stitch, as that made a better, more solid increase. I don’t want this to have any holes for cold air to leak in. For each round where I did increases, I then did a round plain knitting so the hat came together properly.
Once the hat was large enough around, I knit straight until I was ready to start the brim. I did an increase in every stitch to double the total number on the hat. Then, with help from folks at Taos who showed me the technique, I used the tail end of the yarn, knitting from both ends, and double knit the length of the brim. I’ll have to take pictures showing how I did it. Once I figured out how to carry both front and back yarns in one hand without tangling, it went very quickly.

The cast off was done with two stitches: one from each layer, and both yarns at the same time. At the end, I did a simple loop chain to make the loop at the bottom of the cap. This version looks a lot closer to the original. I liked that I could start with precious yarn, knowing that I wouldn’t waste any. I hate long-tail cast-on and other yarn-wasting methods, probably as a result of using my hand-spun a lot of the time. The time and effort I invest into making the yarn means I don’t want to cut any off! Starting at the top means I can knit straight through with no waste. The double knitting went quickly, and eventually made a proper-looking brim easily. The cast-off looks more correct for the bottom than my first version.

There are a few things I’m still not happy with. The increases on top and some of the loose ends look unfinished, as I haven’t finished weaving them in yet. The top increases don’t quite match the original, but the picture of the original is poo, so it’s hard to identify a sure match. I have written the museum to see if there are any better pictures, but none seem to exist. For now, this is my best attempt so far at making a fairly accurate Monmouth cap. At least, it is from very accurate Black Welsh Mountain sheep wool.
*Addendum: The Nelson Museum was eventually renamed the Monmouth Museum, and it permanently closed during the COVID-19 pandemic. A new museum showing the collection will open at the Shire Hall by 2027.