How to keep dust and cat hair off cast iron pan?

I live with three long haired cats that shed in the winter, and my place gets dusty at times too. I have a fairly new cast iron skillet that I use once every one or two days. I dont have any shelves in the kitchen big enough to store it so I have to hang it.

There's always cat hair and stuff stuck to it because I store it on the open. So then I have to clean it which is making seasoning difficult.

How can I store this pan in a way that is both convenient and protects it from the environment? I have found covering it in foil to be somewhat effective but tedious and not perfect, and also wastes a lot of foil. Plastic is difficult because I have to let it cool, although if the oven is off I can let it cool in there. I cant really store it in the oven because i use the oven often. The design of my house makes it impossible to keep cat hair and dust out of the kitchen.

I also dont understand the physics and chemistry behind how the seasoning layers develop so I'm not sure what storage techniques are bad for the condition and "development" of the pan.

Is there a good way to store it that keeps it clean and in good condition?