Tool-augmented large language model (LLM) agents can orchestrate specialist classifiers, segmentation models, and visual question-answering modules to interpret chest X-rays. However, these agents still solve each case in isolation: they fail to accumulate experience across cases, correct recurrent reasoning mistakes, or adapt their tool-use behavior without expensive reinforcement learning. While a radiologist naturally improves with every case, current agents remain static. In this work, we propose Evo-MedAgent, a self-evolving memory module that equips a medical agent with the capacity for inter-case learning at test time. Our memory comprises three complementary stores: (1)~\emph{Retrospective Clinical Episodes} that retrieve problem-solving experiences from similar past cases, (2)~an \emph{Adaptive Procedural Heuristics} bank curating priority-tagged diagnostic rules that evolves via reflection, much like a physician refining their internal criteria, and (3)~a \emph{Tool Reliability Controller} that tracks per-tool trustworthiness. On ChestAgentBench, Evo-MedAgent raises multiple-choice question (MCQ) accuracy from 0.68 to 0.79 on GPT-5-mini, and from 0.76 to 0.87 on Gemini-3 Flash. With a strong base model, evolving memory improves performance more effectively than orchestrating external tools on qualitative diagnostic tasks. Because Evo-MedAgent requires no training, its per-case overhead is bounded by one additional retrieval pass and a single reflection call, making it deployable on top of any frozen model.