Speech summarization is a critical component of spoken content understanding, particularly in the era of rapidly growing spoken and audiovisual data. Recent advances in multi-modal large language models (MLLMs), leveraging the power of LLMs, enable generating textual summaries directly from speech without intermediate transcriptions, while supporting controllable styles and zero-shot generalization. However, open-source MLLMs continue to lag behind the state-of-the-art text-based LLMs, limiting their practical deployment for speech summarization. In this work, we present a novel multi-stage reinforcement learning training framework to enhance the speech summarization capabilities in MLLMs. Our model delivers substantial improvements over strong baselines, outperforms much larger MLLMs, and significantly narrows the gap with state-of-the-art text-based LLMs.