A small-gain theorem for switched interconnected nonlinear systems is set up which provides a tool for analyzing the behavior of switched systems. Also, a method exploiting an average dwell time and the small gain technique is proposed. Based on the small-gain theorem, this technical note investigates the problem of global stabilization of switched nonlinear systems in strict-feedback form whose all zero dynamics are input-to-state stable (ISS) or only some are ISS and others are not. A sufficient condition for the global stabilization problem under a class of switching signals with average dwell time is derived by constructing state-feedback controllers for subsystems. The result is also applied to a class of switched nonlinear network control systems.