I have a new expensive felt hat and have no idea how to add a sweatband or even what to make one out of. I sure don't want to ruin it!

What happened to the sweatband on that expensive Hat

Sweatband (inside the crown) or hatband (outside)? My hat has a sweatband made of linen with a drawstring for adjustment.

It's stitched around the brim edge. I expect it would be a pain to do that without the right machine and some sewing skills.

Another and easier way would be add a band using modern techniques like tape or velcro. There are vendors like sweatbandit.com or headcovers.com that offer such products. But, be advised if you put too thick a band inside the hat you may in effect alter the size of the hat and make it too snug.