Miniature screws are available, but the closest standard size is a number 0, which is available only in 80TPI. The next size, number 1 is available in 64 TPI. It is 0.073" nominal diameter and might be convinced to thread into a 1/16"-64 hole. Alternately, a #1-64 tap could be used to tap the hole oversized. As the difference in diameters is only on the order of 0.008", that would be doable.

Jim H.