Prior research on training grounded factuality classification models to detect hallucinations in large language models (LLMs) has relied on public natural language inference (NLI) data and synthetic data. However, conventional NLI datasets are not well-suited for document-level reasoning, which is critical for detecting LLM hallucinations. Recent approaches to document-level synthetic data generation involve iteratively removing sentences from documents and annotating factuality using LLM-based prompts. While effective, this method is computationally expensive for long documents and limited by the LLM's capabilities. In this work, we analyze the differences between existing synthetic training data used in state-of-the-art models and real LLM output claims. Based on our findings, we propose a novel approach for synthetic data generation, CG2C, that leverages multi-hop reasoning on context graphs extracted from documents. Our fact checker model, FactCG, demonstrates improved performance with more connected reasoning, using the same backbone models. Experiments show it even outperforms GPT-4-o on the LLM-Aggrefact benchmark with much smaller model size.