Large language models (LLMs) have recently shown strong potential for generating project-level unit tests. However, existing state-of-the-art approaches primarily rely on execution-path information to guide prompt construction, which is often insufficient for complex software systems with rich inter-class dependencies, deep call chains, and intricate object initialization requirements. In this paper, we present CAT, a novel call-chain-aware LLM-based test generation approach that explicitly incorporates call-chain and dependency contexts into prompts through dedicated static analysis. To construct executable, semantically valid test contexts, CAT systematically models caller--callee relationships, object constructors, and third-party dependencies, and supports iterative test fixing when generation failures occur. We evaluate CAT on the widely used Defects4J benchmark and on four real-world GitHub projects released after the LLM's cut-off date. The results show that, across projects in Defects4J, CAT improves line and branch coverage by 18.04% and 21.74%, respectively, over the state-of-the-art approach PANTA, while consistently achieving superior performance on post-cutoff real-world projects. An ablation study further demonstrates the importance of call-chain and dependency contexts in CAT.