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Medical agent systems are increasingly expected to support interactive clinical decision making rather than only static question answering. In such settings, effective agents must reuse prior experience across evolving cases, yet existing memory mechanisms often retain raw historical traces that are redundant, noisy, and difficult to govern. More importantly, they rarely distinguish which memories are truly useful for future reasoning. This limits their ability to accumulate compact and reliable experience for long-horizon clinical reasoning. To close this gap, we propose SkeMex, a post-deployment self-evolution framework that improves medical agents through a skill-based memory without updating model weights. SkeMex distills informative interaction trajectories into structured skills that encode reusable procedural knowledge, and organizes them into a multi-branch repository spanning general, task-specific, and action-level experience. To determine which memories should be reused and retained, SkeMex estimates context-dependent utility from environment feedback and uses it to guide value-aware retrieval and repository governance. A closed-loop ``Read--Write--Assess--Govern" lifecycle further supports continual evolution by writing new skills, updating utilities, promoting useful memories, and removing harmful entries. Experiments across diverse clinical tasks show that SkeMex consistently outperforms representative memory-based agents in both offline and online settings. It also generalizes across model backbones and supports transferable skill memory. All data and code will be released publicly.