Large language models (LLMs) in retrieval-augmented generation systems can still produce hallucinations, generating content that is unsupported or contradicted by the source texts and undermines reliability. Recent work addressed this problem by training span-level hallucination detectors using reinforcement learning (RL) and chain-of-thought (CoT) reasoning. In this work, we show through error analysis that incorrect predictions by existing reasoning-based detectors are strongly associated with CoT processes that lack explicit grounding in source evidence, particularly when verification steps do not quote or verify claims against the retrieved documents. This behaviour contrasts with human verification practices in benchmarks such as RAGTruth, where evidence quotation is a prerequisite for determining hallucinated spans. Motivated by this observation, we propose an evidence-grounded RL framework, namely RLSeek, to explicitly enforce active evidence seeking during CoT reasoning by requiring quotation of relevant source segments at each verification step. Experiments on the RAGTruth and NewsSum dataset demonstrate consistent improvements in hallucination span detection performance, with limited additional reasoning overhead and improved robustness in out-of-domain settings.