Large language models (LLMs) are increasingly used for automated data visualization, yet existing approaches often frame visualization generation as a single-step mapping from user query to figure or code, overlooking the iterative analytical reasoning process of expert analysts. We present InsightChain, a four-stage visualization prompting pipeline (Explore--Focus--Test--Present) that emulates expert analytical workflows, together with VG-COPRO, a vision-guided automatic prompt optimization (APO) method adapted to jointly optimize such multi-stage, executable pipelines. To address the evaluation gap for complex data visualization, we introduce the Insight Progression Metric (IPM), a rubric combining four text-based dimensions with a vision-based dimension. We assess IPM through a 100-chain human pilot and an expanded 300-chain agent-based evaluation spanning all ten domains. Experiments on public datasets show that InsightChain consistently outperforms competing prompting baselines. Existing APO methods fail to yield consistent gains on this multi-stage task, whereas VG-COPRO improves performance in both in-domain and cross-domain settings.

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