In an earlier paper, we addressed the problem of stabilization of the rolling sphere about any desired configuration. For the controller proposed, we establish in this paper global stability of the desired configuration and exponential convergence of trajectories to this configuration from a large and well defined set in the configuration space. For configurations that lie outside the set, we define simple maneuvers to move them within the set in finite time such that the sphere can be exponentially converged to the desired configuration thereafter. Our theoretical claims are validated using simulations.