LLM-based recommendation agents personalize what they know through evolving per-user semantic memory, yet how they perform reasoning relies on a universal, static system prompt shared across all users. When a recommendation fails, the agent only updates its memory, but never examines or revises the reasoning that leads to the failure. To close this gap, we propose SAGER (Self-Evolving Agent for Personalized Recommendation), the first framework to equip each user with a dedicated, continuously evolving policy skill. SAGER incorporates three tailored mechanisms for recommendation scenarios: (i) a two-representation skill architecture constrained by a Cognitive Injection Boundary, decoupling a rich evolution substrate from a minimal inference-time injection; (ii) a contrastive differential chain-of-thought engine that diagnoses ranking errors against unchosen near-misses under a refine-not-replace principle; and (iii) skill-augmented listwise reasoning that mitigates the Score-Tie Bottleneck, enabling the slim skill to serve as an active tie-breaker. Evaluated on four public benchmarks, SAGER achieves state-of-the-art accuracy. Furthermore, plugging its policy skill layer into three structurally distinct host agents delivers consistent 9%–12% gains on the H@1 metric. These results indicate that policy skill evolution provides architecture-agnostic improvements that are largely complementary to memory accumulation.