As a pivotal task in data lake management, joinable table discovery has attracted widespread interest. While existing language model-based methods achieve remarkable performance by combining offline column representation learning with online ranking, their design insufficiently accounts for the underlying structural interactions: (1) offline, they directly model tables into isolated or pairwise columns, thereby struggling to capture the rich inter-table and intra-table structural information; and (2) online, they rank candidate columns based solely on query-candidate similarity, ignoring the mutual interactions among the candidates, leading to incoherent result sets. To address these limitations, we propose HyperJoin, a large language model (LLM)-augmented Hypergraph framework for Joinable table discovery. Specifically, we first construct a hypergraph to model tables using both the intra-table hyperedges and the LLM-augmented inter-table hyperedges. Consequently, the task of joinable table discovery is formulated as link prediction on this constructed hypergraph. We then design HIN, a Hierarchical Interaction Network that learns expressive column representations through bidirectional message passing over columns and hyperedges. To strengthen coherence and internal consistency in the result columns, we cast online ranking as a coherence-aware top-k column selection problem. We then introduce a reranking module that leverages a maximum spanning tree algorithm to prune noisy connections and maximize coherence. Experiments demonstrate the superiority of HyperJoin, achieving average improvements of 21.4% (Precision@15) and 17.2% (Recall@15) over the best baseline.