Apply glue all over the underside of the hat brim. Lay the brim glue-side down on the felt and press it firmly to bond. Trim the excess fabric [A]. Cut away the fabric from the head hole.
Measure the hat from one edge of the brim, up over the crown, and down to the far edge. Cut a circle from the red fabric with a diameter equal to this measurement plus
Fill in the area around the crown with bubble packaging to make a toadstool shape, securing it with duct tape [B]. (We made balls to fill in around the brim, then topped them with flat pieces.)
Center the hat upside down on the red fabric circle. Pull the fabric up over the edge of the brim, gluing it as shown to make even gathers [C].
Cut the remaining white felt into 30 to 40 circles, from 1K to 4 inches in diameter. Glue the circles onto the toadstool cap in a random pattern. Have your child try on the cap. If it's too wobbly, glue white-felt strips to the inner hatband until it fits securely.