We define model recovery anti-windup for SISO linear control systems with output saturation. We address the problem by relying on a hybrid modification of the linear closed loop which employs a suitable logic variable to activate/deactivate various components of a control scheme. The scheme relies on a finite-time observation law, an open-loop observer and an open-loop input generator which is capable of driving the plant output within the saturation limits. Then the control scheme is based on suitable (hybrid) resetting laws allowing the controller to operate on the artificial output signal generated by the open-loop observer when the actual plant output is outside the saturation limits. Unlike existing results, not only we prove uniform global asymptotic stability of the closed loop, but we also prove the local preservation and global recovery properties, typical of model recovery anti-windup paradigms. We also illustrate the proposed technique on an example study.