N-ary knowledge graph completion (KGC) aims to infer missing components in facts with multiple entities under distinct semantic roles, commonly formulated as a knowledge hypergraph link prediction task. Most embedding-based approaches score individual hyperedges relying on enriched structural representations, but overlook intermediate propagation states containing complementary local and global structural evidence. Despite their capability to generate chain-of-thought (CoT) representations for the classical KGC task, large language models (LLMs) struggle with hypergraph structure involving multiple facts, while current hypergraph QA methods only provide LLMs with a single query signal rather than path-level evidence. These limitations hinder the transferability of existing methods, especially those leveraging LLMs, to solve the knowledge hypergraph link prediction problem. To bridge this gap, we propose HyperCoT, a structure-aware approach that models multi-hop structural reasoning as a depth-sensitive progressive evidence accumulation process. It constructs a Graphical Chain-of-Thought (Graph-CoT) by aggregating role-aware hyperedge states along strongly correlated reasoning paths, and injects the resulting path-level structural evidence into each token in query and candidate entities to prompt LLMs. Experiments on three real-world datasets demonstrate that HyperCoT consistently outperforms strong n-ary KGC baselines, particularly in high arity and structural sparsity scenarios, meanwhile yielding interpretable multi-hop reasoning traces.