A micrometer will measure string gauges (they read to the thousandth of an inch). They are typically used by macninists and auto mechanics. I have doubts about finding one for $20, but this would be such light-duty use I wouldn't be afraid to buy a real cheapie if one's available.

Precision measuring calipers (essentially the modern equivalent of vernier calipers, but not literally having a vernier-type scale) will also give readings to the thousandth of an inch. This will be either on a dial or an LCD readout. I'd be very surprised if a new one can be had cheaply enough to be tempting.

You might find these used at a garage sale, or by going to machine shops or auto repair shops and asking about.