In 1643, the Oxford Army was issued with coats, breeches and montero caps:
"all the common soldiers then at Oxford were new appareled, some all in red, coats, breeches and mounteers; & some all in blewe"Sadly no one really knows what a montero actually looked like, the Oxford English Dictionary says:
"A cap of a type formerly worn in Spain for hunting, having a spherical crown and (freq. fur lined) flaps able to be drawn down to protect the ears and neck"
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The last two are also similar, one is a Hollar engraving, the other by Abraham Bosse, entitled Envy, both show a peakless cap with piped seams or maybe braided decoration to the crown, which is made of more panels this time, maybe 12 or so and a folded band that comes to a point in front like the drummer's one. The right hand image also shows the folding of the band though I suspect, having made a few, that the artist doesn't quite understand how they work. The detail does show though that the band is butt stitched together at the front. Both have a button on the top that covers the tricky part of the crown where the seams meet.
This second example is made more to the Lunsford pattern with a rounded front to the band
And now a third example, my best take on the peakless Envy montero.