Reference tracking is a challenging problem in the presence of saturation. This paper proposes a hybrid controller that ensures global convergence to zero of the error dynamics of a saturated multidimensional double integrator in the presence of a time-varying reference. Such a controller combines a local linear feedback, which ensures the tracking in the presence of small errors (local mode), and a global stabilizing quasi time-optimal controller, which handles large errors and the saturation (global mode). The proposed switching hybrid logic is applied also to the reference. In global mode, the reference trajectory is slowed down to promote the stabilization task. Once in local mode, the controller speeds up the reference to restore the nominal behavior. The proposed strategy has been extensively tested via simulations. The proposed solution has shown a graceful performance degradation in the presence of disturbances that do not allow the controller to remain in the local mode.