The small tubes of rayon, and even some of the older cones as well, can be a bit tricky to work with. The thread has a tendency to want to come off the tube in one large pile. As a result the thread gets caught underneath the remaining thread on the tube, causing thread breaks.

One technique to avoid this problem is to use a thread coaster or create your own. Similar to a beverage coaster, a thread coaster provides a surface for the thread on the tube to sit on, preventing it from getting trapped underneath, leading to the thread break. Beverage coasters will work just fine, if you can find them, however, you can make your own from some heavyweight backing. Cutaway will work as well, but Tearaway has a stiffer surface. Heavyweight cap backing, like a 2.0 or 3.0oz is the best.
Use a soda can or anIsacord (New Isafil) cone as a template to draw a circle on the backing. Use a pair of scissors to cut out the circle. In the center cut out a hole large enough for the thread retainer on your machine to slide through. Put the coaster on the machine, then add the thread tube. Viola, the thread now runs off smoothly.