Hi, I work a large museum in the U.S. state of Delaware (Not in the conservation department, full disclosure), and a few others- so I think I can help answer your question. Caring for historic objects and artifacts (especially textiles!) can be a serious challenge.

So to follow up, I have a question: Is this slouch hat or forage cap or kepi? These were made of different materials (slouch hats were often made of wool or fur felt, and forage caps were made from wool or various types of weaves). Here's a resource that might help for identifying it.

In either case, I's suggest getting a good acid-free box from a company like Gaylord or Hollinger that the hat will fit inside (acid-free meaning the box is free from the acids usually found in paper products- these acids will break down paper itself and can migrate to objects they come in contact with).

A box like that will protect the hat from it one of the biggest dangers to textiles: UV light. Keeping an object out of direct sunlight is a key first step in preserving it. A box will also create a microclimate to limit the fluctuations in humidity and temperature that can also be damaging to textiles (Great article on that here).

I'd also suggest buying some acid-free tissue to help support the object (a wool forage cap might need a little more support- that's why I ask). Even as an object sits flat, its own weight may be slowly putting tension on the fibers. So, a few rolled or balled-up pieces of acid-free tissue will help support it and keep the stitching and or weave from pulling apart (here's a nice resource with some more details).

Lastly, I'd suggest putting it in a safe space away from potential leaks (either from windows, AC units, or pipes) in your house. The other big risk for a textile is water or mold-growth from high humidity.

As for donating it to a museum, that's your choice. What I've outlined is the steps museums would take with the object. A museum probably will have better climate controlled storage (Hopefully, anyway). That might be the biggest question: Do you live in a house with AC and/ or a way to control humidity? If not, a museum might be a safer place for the object. However, you donating the hat to a museum should be a considered choice. Try to find a museum that has good storage facilities and can use the hat for interpretive purposes and/or exhibits.

I hope this helps! Feel free to reach out to my with further questions- I can put you in touch with actual conservators if you'd like.