Hyperedge prediction plays a central role in hypergraph learning, enabling the inference of high-order relations among multiple entities. However, existing methods often rely on a simplistic \emph{flat set assumption}, treating candidate hyperedges as unstructured collections of nodes and neglecting their potential internal compositionality. Furthermore, the severe scarcity of observed hyperedges poses a challenge for effective supervision. In this work, we propose S$^3$Hyper, a Substructure-contextualized Self-Supervised framework for Hyperedge prediction, which jointly addresses these two challenges. Specifically, we design a substructure-contextualized hyperedge aggregator that models the internal hierarchy of candidate hyperedges by leveraging sub-hyperedge information. In parallel, we introduce an adaptive tri-directional contrastive learning module that incorporates node-level, hyperedge-level, and cross-level alignment objectives, supported by temperature-adaptive mechanisms. Experimental results on four public datasets demonstrate that S$^3$Hyper consistently outperforms strong baselines, with ablation studies verifying the effectiveness of each component.