“It’s a Taz, which applies to today’s hats, which are not what the Mechaber was referring to. “
I saw some Chinese people recently with their Chinese style hats that resemble umbrellas, and I was reminded of this discussion.
The hat that the Gemara (Shabbat 138b)is referring to was called “siana” with a Samech. The Chinese are Sini in Hebrew (also with a Samech).
Based on the description (inflexible, wide brimmed, sun-protective hat) and the name, it seems to me that the hats that the Gemara was referring to could possibly be the Chinese style hats.