This paper deals with the global practical tracking problem by output-feedback for a class of uncertain cascade systems with zero-dynamics and unmeasured states dependent growth. The systems investigated are substantially different from the closely related works, and have zero-dynamics, unknown growth rate, and unknown time-varying control coefficients. This makes the problem much more difficult to solve. Motivated by the authors’ recent works, this paper proposes a new adaptive control scheme to achieve the global practical tracking. It is shown that the designed controller guarantees that the state of the resulting closed-loop system is globally bounded and the tracking error converges to a prescribed arbitrarily small neighborhood of the origin after a finite time. This is achieved by combining the methods of universal control and dead zone with backstepping technique, and using the framework of performance analysis in the closely related works. A numerical example demonstrates the effectiveness of the theoretical results.