Wrap tape around the string just where it passes through the nut if general looseness is suspect.
If it's tight width wise you can shim it higher with just about anything just to take measurements. Thin strip of matchbook cover or business card, whatever.
None of those things will make the string ring true and clear as they are soft items, but they work fine for taking measurements.
The coin thickness thing, or any other thing of the right thickness, wooden matchstick, cut strips of expired debit card, whatever, work to measure the depth of the nut cuts from underneath the strings, laying on the fretboard.
For example, if you can slide whatever it is under 4 of the strings, but have to force it underneath the 5th one (or it just plain don't fit without lifting up the string), you know that nut slot is cut deeper than the others.
