This paper addresses the problem of verifying stability of the interconnection of integral input-to-state stable (iISS) systems. This paper strengthens a Lyapunov approach to the problem to the point where the necessity of a small-gain condition holds without technical assumptions such as uniformity and analyticity required by previous work. This paper unifies the treatment of iISS and input-to-state stable (ISS) systems, which has not been achieved by any previous technique of constructing Lyapunov functions. It is shown that iISS systems and ISS systems can share a common form of Lyapunov functions to characterize stability of their interconnection with respect to external signals. Global asymptotic stability is also discussed.