So I finished embroidering, finished the white silk stuff to go under the pinked other stuff, then I pinned my crown covering to my crown and sewed it down:
Then I sewed my embroidery layer onto the crown, along with all the other stuff:
Meanwhile the underside was this:
The fabric was a silk scarf I bought in error. It’s some sort of pongee and to pleat properly is to attached to a ribbon that you pleat. Everything else is just nonsense. The feel is nice though. Anyway I sewed it to a cotton twill tape, pleated it, attached it to the inner crown and then I folded it over the brim. There is a bias tape under it, protecting it from the edge of the hat.
Then I sewed the outside of the embroidery layer to the resulting edge, without going back through the actual hat. No extra bias tape was necessary. Now the whole thing is just missing some additional decoration and ribbon ties.