No. you misunderstand how gravity is modeled in GR. There is no "space-time fabric height". The distortion of space-time has an maximum in the center of a sphere, causing a maximal time dilation there. Free fallers are always drawn towards areas of higher time dilatation. But at a point of locally maximal time dilatation, there is nowhere else to go.
Try this visualization:
If you set "initial position" to zero, the free falling object is placed at the center of the sphere. And it stays there because it's geodesic world line doesn't deviate in any space direction. The object moves only trough time by moving straight ahead trough space-time.