The open-endedness of large language models (LLMs) combined with their impressive capabilities may lead to new safety issues when being exploited for malicious use. While recent studies primarily focus on probing toxic outputs that can be easily detected with existing toxicity classifiers, we show that LLMs can generate diverse implicit toxic outputs that are exceptionally difficult to detect via simply zero-shot prompting. Moreover, we propose a reinforcement learning (RL) based attacking method to further induce the implicit toxicity in LLMs. Specifically, we optimize the language model with a reward that prefers implicit toxic outputs to explicit toxic and non-toxic ones. Experiments on five widely-adopted toxicity classifiers demonstrate that the attack success rate can be significantly improved through RL fine-tuning. For instance, the RL-finetuned LLaMA-13B model achieves an attack success rate of 90.04% on BAD and 62.85% on Davinci003. Our findings suggest that LLMs pose a significant threat in generating undetectable implicit toxic outputs. We further show that fine-tuning toxicity classifiers on the annotated examples from our attacking method can effectively enhance their ability to detect LLM-generated implicit toxic language.