Where To Put New Finds ?

Wrap in a piece of soft napkin. Or if you carry a handkerchief, wrap in that. Do not place in a piece of paper unless you really know what that paper is made from. If you have a purse, wallet or a tight pocket, place that coin wrapped in something soft and then in there so it doesn't move.

Not really. If you've ever had Chemotherapy you learn to carry a lot of stuff. Your own handerchief that you know is really clean is what is safe considering Chemo lowers your immune system drastically. You don't shake hands, you wipe off door handles, etc. It's all for being safe, not sorry.
Who carries one? Many people after Chemo for one. Try it sometime.

A always have a few old airtites in my car, they're all scratched up but they get the job done.

Pretty easy to put one in your wallet, not sure you can do better than that.

I was kind of picturing that where someone gives you a really valuable coin, you put it there, the phone rings, you open it and out pops that coin now rolling down the street.
I kid picks it up, runs to a vending machine, puts it in and out comes a can of pop now valued at a hundred dollars.
Just kidding but I'll stick to a handkerchief.
But I now have a thought of carrying a 2x2 in my car, just in case.

"Many people after Chemo for one. Try it sometime."

Well now I just feel bad :(

I'm tempted to say I swallow them and retrieve them later but that would be a lie. Either my change pocket like KenRingold just posted or in a pinch when I found many I wanted to keep, I put them in the rim of my sock and rolled it down into a cuff. Worked.

Sock idea, lol, it could work! Just not in the wintertime.
Phone idea is tops - wow, the only downside is I wouldn't be able to send or receive calls for a time.
There's the old rubber band around the shirt trick that I've forgotten about, just gotta remember to carry around rubber bands - you never know!