In practical application scenarios, moving infrared small target detection (MIRSTD) remains highly challenging due to the target's small size, weak intensity, and complex motion pattern. Existing methods typically only model low-order correlations between feature nodes and perform feature extraction and enhancement within a single temporal scale. Although hypergraphs have been widely used for high-order correlation learning, they have received limited attention in MIRSTD. To explore the potential of hypergraphs and enhance multi-timescale feature representation, we propose HyperTea, which integrates global and local temporal perspectives to effectively model high-order spatiotemporal correlations of features. HyperTea consists of three modules: the global temporal enhancement module (GTEM) realizes global temporal context enhancement through semantic aggregation and propagation; the local temporal enhancement module (LTEM) is designed to capture local motion patterns between adjacent frames and then enhance local temporal context; additionally, we further develop a temporal alignment module (TAM) to address potential cross-scale feature misalignment. To our best knowledge, HyperTea is the first work to integrate convolutional neural networks (CNNs), recurrent neural networks (RNNs), and hypergraph neural networks (HGNNs) for MIRSTD, significantly improving detection performance. Experiments on DAUB and IRDST demonstrate its state-of-the-art (SOTA) performance. Our source codes are available at https://github.com/Lurenjia-LRJ/HyperTea.