It pretty much looks the same as before, so no pictures for now. I added the lining and some plain ribbon ties. I am rather pleased with the curve I got in it, though.
It's a regular flat, round straw hat. I marked both the center points and the side points inside the crown with thread. Then I used a hot iron and steam to press curves into the sides. The top silk cover is carefully fitted to the curved shape; it's smooth, almost taut front and back, but fully loose enough on the sides to enable the curve to stay.
The lining, per
reine_de_coudre's suggestion, is a single piece. (No point in separately lining a 1/2" deep crown.) However, I couldn't just sew in the round piece on the edges; it would either be totally tight and unwearable, or it would want to sag like the headliner in an old car. So the first thing I did was to tack it into the crown on top of the front/back and side points. Then I could pin it around the edges, again smoothly to the front and back, and with some tension on the sides. Not enough to make the profile look weird, but enough to help keep the curve. I'm very pleased with it!
I just realized some of the ease of this was using silk broadcloth instead of taffeta. It's the same weight and quite stable, but it was easier to fold on a curve than silk usually is.