I've looked up the Organic Cotton supplier I used to use, they have some interesting products including bamboo silk and cotton Lycra. The Lycra content is 5% so they won't be super stretchy but will tend to cling especially when wet or sweaty.
I've experimented with organic cotton fundoshi wondering if there were a market among those allergic to synthetics and elastics.
https://organiccotton.biz/store/