A standard model of a security game assumes a one-off assault during which the attacker cannot update their strategy even if new actionable insights are gained in the process. In this paper, we propose a version of a security game that takes into account a possibility of a two-phase attack. Specifically, in the first phase, the attacker makes a preliminary move to gain extra information about this particular instance of the game. Based on this information, the attacker chooses an optimal concluding move. We derive a compact-form mixed-integer linear program that computes an optimal strategy of the defender. Our simulation shows that this strategy mitigates serious losses incurred to the defender by a two-phase attack while still protecting well against less sophisticated attackers.