This technical note revisits the problem of designing a static anti-windup gain for enlarging the domain of attraction of the resulting closed-loop system. By utilizing a composite quadratic Lyapunov function, which was originally proposed to study the stabilization problem for linear systems under actuator saturation, an existing LMI based design algorithm is enhanced to result in a nonlinear, possibly continuous, anti-windup gain. This nonlinear anti-windup gain enables us to obtain an estimate of the domain of attraction in the form of the convex hull of a group of ellipsoids, instead of a single ellipsoid that would result from a single Lyapunov function based design. Simulation results demonstrate the features of the proposed design the composite quadratic Lyapunov function brings about.