This article proposes a novel improved adaptive event-triggered (AET) control algorithm for networked Takagi-Sugeno (T-S) fuzzy systems with asynchronous constraints. First, taking the limited bandwidth of the network into consideration, an improved AET mechanism is proposed to save the communication resource. Superior to the existing event-triggered mechanism, the improved AET scheme introduces two adjusting parameters, which further contribute to the economization of the communication resource. Second, with consideration of asynchronous premise variables, a reconstructed approach is applied to synchronize the time scales of membership functions of the fuzzy system and the fuzzy controller. Third, to derive a less conservative sufficient condition for the controller design, a new augmented Lyapunov-Krasovskii functional with event-triggered information and triple integral terms is constructed. Meanwhile, by applying a Bessel-Legendre inequality and extended reciprocally convex matrix inequality together, a new control algorithm is derived with less conservatism. Finally, simulations on a cart-damper-spring system are implemented to evaluate and verify the performance and advantages of the proposed algorithm.