Found and tested:

To achieve authenticity, e.g. when riding western with a cowboy hat, there are protective helmets, which are worn under an oversized hat. In this variant, the hat is simply slipped over the protective shell without a disabled attachment.

The protective shell is available in various sizes, made of polyester composite material with polyester resin. The protective shell does not meet the riding helmet DIN standard. If the wind is strong, the hat will be lost.

Image: hard shell as a base. (https://www.hauptner.ch/de/pferdesport/pferd/westernreiten/zubehoer-western/westernhut-kopfschutz)