I'm using a sheer cotton fabric from the home decor department at JoAnn's. Its cotton content will allow it to press easily without fear of melting. I cut a bias strip that was 30" by 8" - way more than I needed, but you can cut the excess off later. I pressed the pleats the length of the fabric - aiming for about 1/4" - 3/8" pleats. Be very careful not to stretch the fabric at this point. After pleating, fold and press the excess under and trim it off. You'll now have a 30" pleated strip that should be relatively straight. Janet's isn't perfect, so don't sweat any waviness.
Now you want to press a curve into your pleated strip. The bias will do all the work for you. The curved band will make wrapping it on the hat lay smoothly.
These pleats are a little on the narrow side, so I'm going to play around with 3 wider pleats in place of the 4 narrow pleats. The hat is $20 on Amazon (thanks Larry!). It's not as flimsy as Janet's, but it should hold up to shadow-casting abuse well.
Once you get the pleats placed you can trim down off the excess length and add your bow.