The rapid development of large language models (LLMs) gives rise to ethical concerns about their performance, while opening new avenues for developing toxic language detection techniques. However, LLMs' unethical output and their capability of detecting toxicity have primarily been tested on language data that do not demand complex meaning inference, such as the biased associations of 'he' with programmer and 'she' with household. Nowadays toxic language adopts a much more creative range of implicit forms, thanks to advanced censorship. In this study, we collect authentic toxic interactions that evade online censorship and that are verified by human annotators as inference intensive. To evaluate and improve LLMs' reasoning of the authentic implicit toxic language, we propose a new prompting method, Pragmatic Inference Chain (PIC), drawn on interdisciplinary findings from cognitive science and linguistics. The PIC prompting significantly improves the success rate of GPT-4o, Llama-3.1-70B-Instruct, and DeepSeek-v2.5 in identifying implicit toxic language, compared to both direct prompting and Chain-of-Thought. In addition, it also facilitates the models to produce more explicit and coherent reasoning processes, hence can potentially be generalized to other inference-intensive tasks, e.g., understanding humour and metaphors.