Large Language Models (LLMs) like ChatGPT are foundational in various applications due to their extensive knowledge from pre-training and fine-tuning. Despite this, they are prone to generating factual and commonsense errors, raising concerns in critical areas like healthcare, journalism, and education to mislead users. Current methods for evaluating LLMs' veracity are limited by the need for extensive human labor, test data contamination, or limited scope, hindering efficient and effective exposure of errors. To address these challenges, we propose HalluHunter, a novel, fully automated framework for systematically uncovering factual inaccuracies in LLMs. HalluHunter employs a knowledge-graph-based approach, extracting fact triplets to generate diverse question types for single- and multi-hop reasoning using rule-based Natural Language Processing (NLP) techniques. Its iterative process starts with random triplet selection for question generation, followed by adaptive selection in subsequent iterations, targeting triplets where LLMs frequently err based on their performance analysis. Our extensive tests on nine prominent LLMs reveal that HalluHunter can trigger factual errors in up to 55% of tested questions. Moreover, we demonstrate that HalluHunter's test cases, particularly in adaptive selection, could further expose the weaknesses in benchmarking the factuality in LLMs meanwhile maintaining the coverage of questions. All code, data, and results are available at this link: https://github.com/Mysterchan/HalluHunter.