If you rotate the circle sufficiently, it should be possible to show that every point on the smaller circle is, after some amount of rotation, the center of the larger circle. (Or, equally useful, on the edge of the larger circle.)
And then your results for the points at the center (or edge) will apply, yielding the desired result.
So what I'd try, then, is to prove that after rotating the smaller circle along the edge of the outer circle, every point on it is at some point either the center of the large circle, or on the edge of the large circle.
I'm not sure how easy that will be, it's just the immediate solution that screams to me.
Edit: A possible way seems to be using the fact that the rotation is continuous, and that the inner circle rotates through a whole $360$ degrees at least one time as it goes around the outer one. You could probably model this mathematically some way and use the intermediate value theorem or certain results about monotonic functions - it depends on how you'd model it. Sadly, without some sort of function or model, I won't be a ton of help and I don't trust myself particularly well to generate a good one. But if nothing else I feel like this might be a nudge in the right direction, so hopefully it was some help.