Sit the hat the right way round and not upside down when in its box. It keeps its shape better that way.
Use plenty of tissue and make sure that the tissue is supporting the hat. If the brim is a down-brim the put lots of tissue in the crown of the hat so that it is sitting on that tissue and not on the brim that could otherwise go out of shape.
Likewise with a small hat with a veil, make sure the veil sits over a ball of tissue so that it does not lose its shape. This is especially relevant to veils that have been blocked (shaped) which we always do.
Thank you so much Rachel! It seems the equation for successful hat storage is simple: store the hat upright in a box entirely supported with tissue. I suppose if you have many hats, a photo or description on the box would be helpful for organization as well. I imagine Queen Elizabeth’s hat closet is a large set of shelves filled with hundreds of hat boxes in varying sizes, perhaps organized by colour. Do you have other handy tips for hat storage you can share?