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You can buy effective faraday pouches for car keys in many places. I have one, and can attest they work. They are no thicker than any other fabric.

I've not tried it for my phone though.

Wouldn't a metal biscuit tin work? You can test them by putting your smartphone inside, and then try to call the phone. Or pinging its IP address...

A plaster walled room (on grate backing) with lead paint and iron wrought window cages is also extremely effective at blocking RF.

Kitchen aluminum foil.