Manager Explicitly Disallowing Me From Wearing The Hat I Use To Protect Myself From The Sun Outside

Location: Texas

I work for a big box retailer and our company recently got a new CEO who's decided he's going all in on an extremely strict interpretation of our dress code.

One side effect of this is that my manager has decided that the style of hat I've been wearing during the summer to protect myself from the sun for around a decade with no problems is suddenly unacceptable to wear even outside. He's completely unwilling to compromise on this(I floated the idea of getting a doctor's note but my supervisor said it would be unlikely to change our manager's mind) and even extends the dress code to when I'm in the store but off the clock. We have company branded ballcaps but they're woefully inadequate at any sort of sun protection.

Summer in Texas is absolutely brutal, it frequently gets above 100 degrees and stays there for weeks at a stretch. I'm less heat tolerant than most people and I have a family history of skin cancer on my mom's side so you can imagine this is causing a fair amount of friction.

Do I have any recourse here? Texas is an at-will employment state so I could be fired if I escalate this, but at the same time I don't want to end up saddled with a massive hospital bill because my boss took issue with a piece of cloth I wear to make sure I don't get heatstroke.

UPDATE: I filed an OSHA complaint. Not expecting a ton to come of it given how Texas is, but hopefully I can make a positive change. That said, I am concerned about possible "you can't prove it's retaliation" that companies like this sometimes do to people that stick up for themselves.