Job scheduling is a critical component of workload management systems that can significantly influence system performance, e.g., in HPC clusters. The scheduling objectives are often mixed, such as maximizing resource utilization and minimizing job waiting time. An increasing number of researchers are moving from heuristic-based approaches to Deep Reinforcement Learning approaches in order to optimize scheduling objectives. However, the job scheduler's state space is partially observable to a DRL-based agent because the job queue is practically unbounded. The agent's observation of the state space is constant in size since the input size of the neural networks is predefined. All existing solutions to this problem intuitively allow the agent to observe a fixed window size of jobs at the head of the job queue. In our research, we have seen that such an approach can lead to "window staleness" where the window becomes full of jobs that can not be scheduled until the cluster has completed sufficient work. In this paper, we propose a novel general technique that we call \emph{split window}, which allows the agent to observe both the head \emph{and tail} of the queue. With this technique, the agent can observe all arriving jobs at least once, which completely eliminates the window staleness problem. By leveraging the split window, the agent can significantly reduce the average job waiting time and average queue length, alternatively allowing the use of much smaller windows and, therefore, faster training times. We show a range of simulation results using HPC job scheduling trace data that supports the effectiveness of our technique.