This paper studies the output regulation of nonlinear systems using conditional servocompensators. Previous work introduced the conditional servocompensator that acts as a traditional servocompensator in a neighborhood of the zero-error manifold, while acting as a stable system outside a boundary layer, leading to improvement in the transient response while achieving zero steady-state tracking error in the presence of time-varying exogenous signals. The conditional servocompensator tool was introduced for sliding-mode feedback controllers. This paper extends the technique to more general feedback controllers by using Lyapunov redesign and saturated high-gain feedback.