Although harder to find there are steering wheel covers out there that aren't so thick. They are made cheaper but cost less. So either find a thinner steering wheel cover or get used to the extra thickness. I use a steering wheel cover on my car and at first the extra thickness bothered me but I quickly got used to it. Get one that matches your cars steering wheel or interior color and one with a good grip.

Wheelskins, real stitch on leather is the name of the game.

we used to get a wrap for the wheel that had a small plastic cord to go around the wrap but i have not seen one in a while now

I have Wheelskins on some of my cars. Very nice leather and not bad looking. It actually look better over time as it conforms.

https://www.wheelskins.com/

https://www.google.com/search?q=wheelski...

Yeah, I'll send them some business, but then they make a very good product.

buy a steering wheel from a wreckers

My steering wheel is deteriorating and I need a way to cover it up, but every steering wheel cover I have tried increases the thickness of the wheel by a good amount, which I find annoying. Is there any kind of thin wrapping or tape I can use instead of a cover? I keep hearing that many tape adhesives will melt in the heat and make a mess, so I'm very hesitant to try anything that isn't made for this specific purpose.