Material that is being cut without the mat (such as vinyl) is loaded into the Silhouette properly, but during the cutting process is starting to slip out of the grip of the rollers thus causing the material to drift and not be cut properly.
This issue is the result of a loading problem where the material was not being properly gripped evenly and thus began drifting during the cutting process. In this instance, while the material may appear to load correctly, it is most often actually being fed in at a slight angle so that over time the pinch rollers are not fully gripping one side. At that point, the material is coming loose and falling off track.
Similarly, one side of the material may be gripped properly, but the other side (while initially appearing so) is not truly being gripped.
To correct this concern, check the following points:
rather than simply a straight cut that is actually at a slight angle in relation to the sides). This way the material is not being fed into the Silhouette at a slight (and generally imperceptible) angle.
Even if the leading edge looks perfectly straight, it may be at an angle in relation to the sides. If the you have access to a paper trimmer, you may wish to align the side of the material with the paper cutter and slice off the leading edge. This should provide a cut that is perpendicular to the edges.
Proper even grip alignment should further resolve the concern as the material is fed in straight.
Place the material into the Silhouette (using the precautions noted above), and then prior to cutting, go to the Silhouette Cut Settings panel. In this screen, use the blue position arrows (which are usually reserved for performing a Test Cut) to feed the material into the Silhouette. To do so, hold down the forward arrow. This will move the material forward and allow the material to be sent through without cutting so that it can be verified that it will go through the machine properly. When finished, simply use the opposite arrow to feed the material back to the starting end and proceed to cut the job.