This paper addresses the problem of establishing stability of interconnected nonlinear systems consisting of individually input-to-state stable (ISS) subsystems. This paper aims to extend the capability of the ISS small-gain technique with emphasis on Lyapunov functions and necessity. This paper derives an explicit formula for constructing Lyapunov functions characterizing ISS property of interconnected systems from dissipation inequalities of subsystems. This paper focuses on the case where nonlinear loop gain approaches unity asymptotically as magnitude of signals tends to zero and infinity. The new formula of Lyapunov functions deals with the non-uniform contraction of loop gain and covers one of authors' previous developments as a special case. The Lyapunov functions obtained here are differentiable. The formulation based on dissipation inequalities also allows us to prove the necessity of the small-gain condition without making a technical assumption of uniformity.