When it comes to travelling, hats are an essential item. The problem is, if you take an ugly hat that is easy to pack, you end up with all your photos being of your ugly hat. So the goal is to bring a beautiful hat and make sure it arrives at your destination in perfect condition.
There are three simple options :
1. Wear your hat on your head - they are fantastic for sleeping on planes because you can place it over your face, shade your eyes, hide your drooling and can see what is going around you. Plus you look like a Mexican gangster. Often I will place my hat on top of my luggage after I take off, and it will still be there when I arrive. Personally I love seeing everyone at the freezing cold Melbourne airport, wearing their hats and Hawaiin shirts, off on a big adventure. I often see my hats and boxes, and it feels good.
2. Put it in a hatbox (or any box really). You can then place that inside your luggage, and fill the box with light items that won't crush your hat (t-shirts, cashmere jumpers, silk underwear etc.). If you have a hatbox, then you can pack two or three hats in the same space. I usually pack all the family hats in my box. You can then use it as carry-on, or pack it in your bag. If your hat is a fino or very soft, then always use a box or wear it.
3. Pack your hat carefully in your luggage. Place it flat on the base of your bag, fill the crown (where your head goes) with soft things, and pack around it gently. Taking care to see then brim is sitting as flat as possible. If it gets a little crushed, you can steam in a bathroom, where it will absorb moisture and let it come back to shape.
The main idea is to have a plan, and take your time to care for your hat. Personally I like to travel with three hats, one to wear, one as a backup, and one I can give away to someone I like.