The leading approach to solving large imperfect information games is to pre-calculate an approximate solution using a simplified abstraction of the full game; that solution is then used to play the original, full-scale game. The abstraction step is necessitated by the size of the game tree. However, as the original game progresses, the remaining portion of the tree (the subgame) becomes smaller. An appealing idea is to use the simplified abstraction to play the early parts of the game and then, once the subgame becomes tractable, to calculate a solution using a finer-grained abstraction in real time, creating a combined final strategy. While this approach is straightforward for perfect information games, it is a much more complex problem for imperfect information games. If the subgame is solved locally, the opponent can alter his play in prior to this subgame to exploit our combined strategy. To prevent this, we introduce the notion of subgame margin, a simple value with appealing properties. If any best response reaches the subgame, the improvement of exploitability of the combined strategy is (at least) proportional to the subgame margin. This motivates subgame refinements resulting in large positive margins. Unfortunately, current techniques either neglect subgame margin (potentially leading to a large negative subgame margin and drastically more exploitable strategies), or guarantee only non-negative subgame margin (possibly producing the original, unrefined strategy, even if much stronger strategies are possible). Our technique remedies this problem by maximizing the subgame margin and is guaranteed to find the optimal solution. We evaluate our technique using one of the top participants of the AAAI-14 Computer Poker Competition, the leading playground for agents in imperfect information setting