<p>Global adaptive control of uncertain parametric systems with mismatched nonlinearities and quantized states is a challenging problem due to the discontinuity of feedback signals. In the existing relevant results, quantization errors are typically required to be bounded by a constant, while system nonlinearities must satisfy the global Lipschitz continuity condition. To relax the constraints, this paper develops a non-backstepping global adaptive quantized feedback control approach by combining a novel time-varying matrix inequality with switching logic. This approach is applicable to more general mismatched nonlinear systems and to coarser logarithmic-type quantizers. It is shown that, as long as the quantization dead-zone size is set sufficiently small, the closed-loop system states can be steered to an arbitrarily small neighborhood of the origin, and the switching-type gain remains bounded. Moreover, an improved logarithmic quantizer is proposed to mitigate the chattering effect that may arise in a conventional logarithmic quantizer. Finally, simulation examples and comparisons are conducted to demonstrate the effectiveness and advantages of the control approach proposed in this paper.</p>