The second-order sliding mode control generates important properties for closed-loop systems, such as robustness, disturbance rejection and finite-time convergence. In this study, it is shown that the adding a power technique plus the nested saturation method will bring in a new second-order sliding mode control scheme for non-linear systems with relative degree two. Based on this, a second-order sliding mode controller is constructed by imposing a natural assumption on the sliding mode dynamics, that is, the uncertainty of the sliding mode dynamics can be bounded by a known function instead of a constant. Under the proposed sliding mode controller, it is proved that the closed-loop system is not only globally convergent, but also locally finite-time stable, which implies the global finite-time stability. Finally, the effectiveness of the proposed method is verified by a numerical example.