High-Order Sliding Mode (HOSM) control was originally proposed to overcome the dangerous chattering effect. The idea is to treat the time derivative of the actual control as a new control artificially raising the relative degree of the sliding variable. The resulting HOSM features finite-time stability, ultimate accuracy of sliding mode, and smooth control. Unfortunately, due to the interaction between the control and its derivative, the convergence to the HOSM is only ensured, if the initial values of the successive sliding-variable derivatives are small enough. It is proved in the paper that under mild conditions that restriction is removed. Output-feedback controllers are constructed. Computer simulation confirms the applicability of the approach.