There are any number of things people use for rigging. I use several different things myself, depending on the size I want to use. For larger size, I use plain sewing threads. The craft stores tell me sewing is a hobby in decline, which is why they no longer carry as big a selection as they used to. Model Expo, a modeling mail order place, carries the finest conventional thread of any place I know.
For medium size line, I use nylon monofilament (including the translucent stuff). For really Fine stuff I use fly fishing thread from a fly tying and fishing place. They have stuff as fine as 5 mil.
Can you confirm the size of the stuff you are using- .001 millimeter? That is thinner than human hair by a large factor. In fact, one micron would be virtually invisible to human eye.
Only bad thing about the monofilament thread I use for medium lines is that it is stiff, very hard to knot. It is like trying to knot fine stainless steel wire!