In this paper, we revisit the conservatism of gain-scheduled control design under common Lyapunov functions Although recent research tends to seek parameter-dependent Lyapunov functions to reduce the conservatism, we point out that the conservatism arising from seeking a common Lyapunov function can be reduced in a different manner to the conventional method. If a condition is satisfied, we obtain a set of extreme controllers that achieve the best performance at vertices. Otherwise, an interpolated controller can be constructed via an interesting combined convex structure. An illustrated example demonstrates the applicability of the proposed method.