Pre-trained Large Language Models (LLMs) require post-training methods such as supervised fine-tuning (SFT) on instruction-response pairs to enable instruction following. However, this process can potentially harm existing capabilities learned during pre-training. In this paper, we investigate the loss of context awareness after SFT, where context awareness is defined as the ability to extract and understand information from user-provided context and respond accordingly. We identify and demonstrate that the loss of context awareness, particularly in open-source models, occurs in instruction fine-tuned LLMs when the chat template is applied to input prompts. We identify that the performance decline is associated with a bias toward different roles learned during conversational instruction fine-tuning. We demonstrate this correlation by visualizing changes in attention allocation after the chat template is applied and manually steering the attention heads. The bias can be learned from training examples that align with the model's internal knowledge and rely less on the user-provided context to generate correct responses. Based on these observations, we propose a metric to identify context-dependent examples from general instruction fine-tuning datasets. We then apply conditional instruction fine-tuning with a context-dependency indicator, enabling the model to preserve context awareness after SFT. Empirical experiments on four context-dependent downstream tasks and three pre-trained LLMs of different sizes show that our method effectively mitigates the loss of context awareness without compromising general instruction-following capabilities.