I store my hats in cardboard boxes with clear windows (that I got at IKEA) and one of those old fashioned round suitcases (that I bought on eBay). I know bonnets are an odd shape, but IKEA has a lot of sort of tall, rectangular storage boxes that might suit. Also I've got a few actual hatboxes that I've grabbed over the years at places like Michaels, but the really big ones are super hard to find.