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We consider control systems for which we know two stabilizing output feedback controllers. One is globally asymptotically stabilizing, while the other one is only locally asymptotically stabilizing. We look for a composite output feedback control law that is equal to the local feedback on a neighborhood of the origin and that is globally asymptotically stabilizing. Since we want some robustness with respect to measurement noise, actuator errors, and external disturbances, we need to consider hybrid output feedback controllers. Under an input-output-to-state stability assumption, we exhibit a solution of this uniting problem by means of a dynamic hybrid output feedback controller. Then, we particularize our study to linear control systems with saturation at the input for which we know two stabilizing output feedback controllers. One is a nonlinear globally asymptotically stabilizing controller, while the other one is a high-performance linear-only locally asymptotically stabilizing controller. We specify numerically tractable conditions to solve this uniting problem. Finally, we illustrate our main results by means of numerical examples.