I solved this problem once by buying the exact cap, and sewing on a patch. If patches are available, and you have the patience for it, it works great!
YP Classics Retro Trucker Cap https://www.amazon.com/dp/B07BR973C5
This is what we use.
Cascade Mountain Tech Camping Chair - Low Profile Folding Chair for Camping, Beach, Picnic, Barbeques, Sporting Event with Carry Bag https://www.amazon.com/dp/B07DR61XDD/ref=cm_sw_r_cp_api_glt_fabc_FQN8TTAQ2A9CC5ACYPJD?_encoding=UTF8&psc=1