Having tried to put on the head wrap about half a dozen times and looked these photographs over, I have a number of observations that may be worth thinking about:
1. The wool band I used is probably a bit wider than necessary, and is too wide for the resulting style to look like the images on ancient Greek pottery. It's about an inch and a half (roughly 4 cm) wide, and it should probably be no more than an inch wide (2.5 cm) to resemble the period images. That argues for using a tablet woven band or a commercially woven wool tape in an appropriate size.
2. It is critical that the tape be wool and that the fabric be linen or a cotton rough enough not to be slippery. Using different fibers would make it likely that the wrap would slide off one's head, no matter how tightly it was tied. The fact that wool is easily dyeable with period-available substances enhances the ornamental value of such a wrap, and further supports the use of wool for the purpose.
4. It's difficult to wrap the band tightly if you are putting the head wrap on your own hair, because it's difficult to flip the ends of the band around in a way that guarantees that the entire length of the band will continue to lie smoothly while you are flipping them. That leads me to the conclusion that Greek women probably used a cloth bag with a band attached for this style, instead of a flat piece of cloth. That way, it would be easier to keep the bun covered as you wrap the band.
5. Once the wool is knotted, however, the wrap is quite stable even if the band isn't knotted tightly, and it tends to stay put even if the band is not double-knotted. The stability is a tribute to the self-sticking qualities of wool, and shows that this would have been a very practical style for a busy Greek woman.
If I make another such wrap, I will use a thin, narrow, tablet-woven band instead of thick wool. I believe the resulting wrap would look both more authentic and more beautiful that way. Still, this was a quick, interesting, and educational project, and I'm glad I did it.