I've had some amazingly goods results from a modern, miniature 220uH inductor (one of the ones that looks just like a resistor).
Placing it on its side on plain steel, the measured inductance goes up to nearly 260uH, then sliding in pieces of 200um thick plastic sheet between the metal and inductor, the measured L falls back down again, useful out to at least 600um. It's quite insensitive to the presence of my hand. I'd say it probably gives me a resolution of ~10um.
The reason I want a gauge is for use in machine-polishing a 20 year old car, which has lots of swirl marks. You really do not want to polish through the clear-coat. You can only measure the clear-coat thickness using infrared, but those instruments are very, very expensive. So, I'm just probably looking at just setting myself an absolute limit of not polishing off any more than maybe 50um (and ideally, less), which should mean I'm still in the clear-coat.