Single-domain generalization (SDG) in time series classification (TSC) poses significant challenges for current time-series domain generalization methods due to the extremely limited data available from only one source domain. In this study, we propose Segment-dErived Expansion of Domains (SEED), a diffusion-based method that effectively expands domain diversity for SDG. We reveal that individual instances exhibit intrinsic temporal shifts over time, which provides a principled foundation for creating multiple pseudo domains by segmenting each instance into distinct parts. To do so, SEED extracts two complementary representations from each time-series segment: 1) a segment-specific representation that captures diverse distributional variations, and 2) a segment-invariant representation that preserves class semantics. SEED formulates these representations as pseudo-domain prompts to guide a diffusion model in generating diverse yet semantically consistent time-series data. Additionally, SEED introduces a novel prompt-fused sampling for diffusion, enabling flexible recombination of segment-specific features to continuously expand the pseudo-domain space. We provide both theoretical analysis and extensive empirical evaluations on four widely used TSC benchmarks to validate its ability in reducing generalization error and improving model's performances in SDG. In our experiments, SEED significantly improves classification accuracy by 7.68% on average compared to the strong baselines.