In this paper we report on further developments on a novel scheme for the anti-windup compensation of rate and magnitude saturated systems. This scheme, originally proposed in [10] allows to rely on tools for anti-windup compensation for magnitude saturated systems also for the case of rate saturation, by way of a suitable model of the magnitude and rate saturation. The novelty in this paper is to show that, for performance improvement, the compensation scheme of [10] can be modified using a saturated integrator for the magnitude saturation part and only retaining the anti-windup gain for the rate saturation part. This modified strategy is shown mathematically to ensure stability and, on a simulation example, it is shown to induce more desirable closed-loop responses.