Traffic signal control is crucial for managing congestion in urban networks. Existing myopic pressure-based control methods focus only on immediate upstream links, leading to suboptimal green time allocation and increased network delays. Effective signal control, however, inherently requires a broader spatial scope, as traffic conditions further upstream can significantly impact traffic at the current location. This paper introduces a novel concept based on the Markov chain theory, namely multi-hop upstream pressure, that generalizes the conventional pressure to account for traffic conditions beyond the immediate upstream links. This farsighted and compact metric informs the deep reinforcement learning agent to preemptively clear the present queues, guiding the agent to optimize signal timings with a broader spatial awareness. Simulations on synthetic and realistic (Toronto) scenarios demonstrate controllers utilizing multi-hop upstream pressure significantly reduce overall network delay by prioritizing traffic movements based on a broader understanding of upstream congestion.