Many recent works on the stabilization of nonlinear systems target the case of locally stabilizing an unstable steady‐state solution against small perturbations. In this work, we explicitly address the goal of driving a system into a nonattractive steady state starting from a well‐developed state for which the linearization‐based local approaches will not work. Considering extended linearizations or state‐dependent coefficient representations of nonlinear systems, we develop sufficient conditions for the stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. On the basis of these analytical results, we propose a scheme that is designed to maintain the stabilization property of a Riccati‐based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in 2 examples.