Pretrained 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 pretraining. In this paper, we investigate the loss of context awareness after SFT, defined as the capability to extract and understand information from the user-provided context and respond accordingly. We are the first to identify and show that the loss of context-awareness appears on instruction-finetuned LLMs when the chat template is applied to the input prompts. We identify the performance decline is partially caused by the bias embedded into the chat template to focus less on the user-provided context. Based on these observations, we propose two methods to mitigate the loss of context awareness in instruct models: post-hoc attention steering on user prompts and conditional instruction fine-tuning with a context-dependency indicator. Empirical experiments on 4 context-dependent downstream tasks and 3 pretrained LLMs of different sizes show that our methods effectively mitigates the loss of context awareness without compromising the general ability to follow instructions. Our findings also strongly advocate the necessity to carefully benchmark context awareness after instruction fine-tuning.