Continuous dimensional speech emotion recognition captures affective variation along valence, arousal, and dominance, providing finer-grained representations than categorical approaches. Yet most multimodal methods rely solely on global transcripts, leading to two limitations: (1) all words are treated equally, overlooking that emphasis on different parts of a sentence can shift emotional meaning; (2) only surface lexical content is represented, lacking higher-level interpretive cues. To overcome these issues, we propose MSF-SER (Multi-granularity Semantic Fusion for Speech Emotion Recognition), which augments acoustic features with three complementary levels of textual semantics--Local Emphasized Semantics (LES), Global Semantics (GS), and Extended Semantics (ES). These are integrated via an intra-modal gated fusion and a cross-modal FiLM-modulated lightweight Mixture-of-Experts (FM-MOE). Experiments on MSP-Podcast and IEMOCAP show that MSF-SER consistently improves dimensional prediction, demonstrating the effectiveness of enriched semantic fusion for SER.