One of the particularly daunting issues in the cybersecurity domain is information leakage of business or consumer data, which is often triggered by multi-stage attacks and advanced persistent threats. While the technical community is working on improved system designs to prevent and mitigate such attacks, a significant residual risk remains that attacks succeed and may not even be detected, i.e., they are stealthy. Our objective is to inform security policy design for the mitigation of stealthy information leakage attacks. Such a policy mechanism advises system owners on the optimal timing to reset defense mechanisms, e.g., changing cryptographic keys or passwords, reinstalling systems, installing new patches, or reassigning security staff. We follow a game-theoretic approach and propose a model titled FlipLeakage. In our proposed model, an attacker will incrementally and stealthily take ownership of a resource (e.g., similar to advanced persistent threats). While her final objective is a complete compromise of the system, she may derive some utility during the preliminary phases of the attack. The defender can take a costly recovery move and has to decide on its optimal timing. Our focus is on the scenario when the defender can only partially eliminate the foothold of the attacker in the system. Further, the defender cannot undo any information leakage that has already taken place during an attack. We derive optimal strategies for the agents in FlipLeakage and present numerical analyses and graphical visualizations.