For the green panel on the front of the hat, either a stiff fabric or a fabric that has been lined with stiff interfacing will hold its shape like that.
For the hat itself, I would go with a bucket hat (tutorial) and exclude the brim. The structure of the hat and a double layer of fabric will help support its shape. However, you can also use a lightweight interfacing or additional layer of fabric for added structure or bulk.
Hope this helps!