Second- and higher-order sliding mode techniques are able to avoid the chattering effect associated with first-order sliding. The super-twisting algorithm is one of the most popular second-order sliding mode techniques. Existing modifications of the super-twisting algorithm allow for improved disturbance rejection capability. In this paper, we introduce a new class of generalizations of the super-twisting algorithm that are able to keep the main advantages of standard super-twisting while rejecting disturbances bounded in forms for which the existing algorithms may be not directly applicable. Our results thus broaden the applicability of second-order sliding-mode techniques. We give conditions for stability and finite-time convergence, and provide a complete design framework based on given disturbance bounds.