In my delightful forays into online museum collections, I found a piece of classical Greek pottery decorated with a closeup of a woman's head clad in a cap. It is British Museum no. 1756,0101.485.
The cap's structure looks similar in at least three ways to some styles of extant Coptic språng caps that were made a thousand years after the pot was painted.
A quick look at thumbnails for the same search results shows there are more pots like this. It's exciting. Most everything I've seen to date has shown figures head to toe not close up, and as often as not with the cap-clad head positioned at a spot on the pot that curves away and can't be seen easily.
It's exciting not so much because this information will help me make a cap. It's more that I am starting from a premise that the Greeks used språng for headgear and I'm out to find primary evidence. The premise comes from research-driven authorities on språng construction and ancient Greek culture, for example, Peter Collingwood in The Techniques of Sprang, Margrethe Hald in Old Danish Textiles, Elizabeth Siewertsz van Reesema in her works including the article "Old Egyptian Lace," and Marina Fischer in her thesis, "The Prostitute and Her Headdress: the Mitra, Sakkos and Kekryphalos in Attic Red-figure Vase-painting ca. 550-450 BCE." I like to compare and contrast things to draw conclusions. In this case, a Greek picture compared with a Coptic piece. I'm following clues for something is a bit of a mystery. It wouldn't be if we'd had an unbroken widespread tradition of språng.
If and when I do plan out the specs for a cap, I have dimensions for quite a few existing Coptic pieces and I think I can estimate the dimensions of the looms on a few pieces of Greek pottery.