How about a cat carrier for sleeping and take her out to the car in it in the morning?

I’ve traveled with two cats and a dog cross country two times. I realize it’s no fun. Cats hate being in the car. They know that they’re free at night to roam the room, but that in the morning they will have to be put back in the carrier and into the car.

Buy a medium sized dog crate. Set it up with her food, water and a small disposable litter box. Leave the crate door open. When it’s bedtime put her in the crate and shut the door. If you have an extra blanket or sheet cover the crate with it. In the morning you can get her out and put her in her cat carrier.

Hope this helps.

I would put a harness on the cat with a leash, so in case the cat takes off, it will be easier to catch if it has a leash trailing than if it’s got nothing. Also, you can use the leash to tether the cat into the car but it won’t hang itself with a harness in lieu of a collar (of course the cat should have a collar on, but the leash will be attached to a harness, not the collar).

Also, put ID on the cat’s collar, on a tag – cell number, etc.

You can travel more often until the cat adapts, but if travel is to be routine, be certain your next cat is a kitten who can grow up a road kitty. I know a couple of different people with cats who live virtually on the road to earn a living. Their cats apparently love it and race to the car in the mornings with enthusiasm.

There are also calming supplements that can help your cat. I’ve tried Bach Flower remedies and they work well. You can put it in their water. It’s in the link below, along with quite a few other options:
https://www.thesprucepets.com/natural-cat-stress-remedies-555543

I agree with stanleymanly

In that to get a kitten and rear it to accept the quick changes in environments of travelling.

From a cats perspective its torn from its roots every time it adjusts to its environment then off again?

That cat is lower to the ground and can smell, sense, see things that we take for granted. It must be terrorized?

I grew up with cats and looked after a neighbors cat that did the same thing as yours did.

Exploring its surroundings, hiding under furniture .

It took about a week for it to realize that there was nothing to worry about and then it relaxed .
I ended up having that cat for a few months before I realized that it was dumped on me under false pretenses.

Speaking of calming agents for cats, you can also buy pheromone calming spray. You spray it in the carrier, crate or even your car. It has no detectable odor to humans.