Smartphones have become essential to people's digital lives, providing a continuous stream of information and connectivity. However, this constant flow can lead to moments where users are simply passing time rather than engaging meaningfully. This underscores the importance of developing methods to identify these "time-killing" moments, enabling the delivery of important notifications in a way that minimizes interruptions and enhances user engagement. Recent work has utilized screenshots taken every 5 seconds to detect time-killing activities on smartphones. However, this method often misses to capture phone usage between intervals. We demonstrate that up to 50% of time-killing instances go undetected using screenshots, leading to substantial gaps in understanding user behavior. To address this limitation, we propose a method called ScreenTK that detects time-killing moments by leveraging continuous screen text monitoring and on-device large language models (LLMs). Screen text contains more comprehensive information than screenshots and allows LLMs to summarize detailed phone usage. To verify our framework, we conducted experiments with six participants, capturing 1,034 records of different time-killing moments. Initial results show that our framework outperforms state-of-the-art solutions by 38% in our case study.