This is super easy to do if you have a sewing machine with a button hole stitch and foot. First, follow all the steps from the original tutorial.
Next, you need to add a button hole on each side of the brim. Measure 3/8" away from the base of the crown and make a mark. (You want the button holes to be perpendicular to the seams in the brim.) Choose a button that will make a button hole slightly longer than the width of the chin straps. Sew the button holes and cut out the fabric inside.
When you want to reverse the hat simply pull the chin straps through the button holes.