You'll want a design that can wrap around the sides a little and something simple for the top.

I suggest creating your own wrap as any preset designs probably won't fit the curve of your hat and can be hard to work with.

Flowers are a quick go to but you can also use more words, images styled to a music festival, etc.

For the top I picked a short word that would be clear to read.

Larger areas of engrave (like the flower centers) are more prone to burning so I suggest keeping them to a minimum.

You can use a couple of binder clips to hold the hat down on a 12x12 sheet of plywood.  

This will help to hold it flat for a more consistent focus level.  Put the brim under the laser lights to align them and then you can move it.  At an angle in the machine you can get the entire side in one go.

Use the framing outline to precisely frame your design and where it will land.

Felt hats are prone to flames so do not leave your machine alone for a second while working on this.

You won't be able to close the machine so set up a shield to protect everyone's eyes and use your eye protection.

On my hat I found that the top and the sides needed different settings so you'll probably want to sacrifice a hat from the bulk purchase on the test grids.  Test on the brim AND on the top.

Remember that you can also engrave the underside of the brim.

Or even the inside of the top if you want your name labeled on it.

Just remember to refocus every time you turn it in a new direction.