The very best flats shirts have the multi-level buttons to allow a 'turtle neck' style button up but only one or two models of hats have the flaps that button and these flaps are not always on the best hats.
So in searching for a solution I stumbled upon elastic terry cloth headbands at the sporting goods store for $2.99.
Simply button the shirt to the top, flip the collar up. Then put on your flats hat. Pull the headband over the flaps and collar and voila.
Helps on very hot days as well to contain sweat.
Now to dye them into stealh sky colors...