The trick to keeping your hat in tip-top shape is support and dryness. Strain and moisture will cause your hat to wilt. So don't store your hat with all of it's weight resting on the brim or tail.
After testing all kinds of methods for storing hats so they keep their shape I now store my hats by hanging them. This does not put pressure on the brim and it allows gravity to help the hat tail keep its shape. (see photo)
Don't store your hat in a place with high humidity such as a bathroom or basement. It may help to run a dehumidifier nearby if you typically have very humid weather.