In large-scale cloud service platforms, thousands of customer tickets are generated daily and are typically handled through on-call dialogues. This high volume of on-call interactions imposes a substantial workload on human support analysts. Recent studies have explored reactive agents that leverage large language models as a first line of support to interact with customers directly and resolve issues. However, when issues remain unresolved and are escalated to human support, these agents are typically disengaged. As a result, they cannot assist with follow-up inquiries, track resolution progress, or learn from the cases they fail to address. In this paper, we introduce Vigil, a novel proactive agent system designed to operate throughout the entire on-call life-cycle. Unlike reactive agents, Vigil focuses on providing assistance during the phase in which human support is already involved. It integrates into the dialogue between the customer and the analyst, proactively offering assistance without explicit user invocation. Moreover, Vigil incorporates a continuous self-improvement mechanism that extracts knowledge from human-resolved cases to autonomously update its capabilities. Vigil has been deployed on Volcano Engine, ByteDance's cloud platform, for over ten months, and comprehensive evaluations based on this deployment demonstrate its effectiveness and practicality. The open source version of this work is publicly available at https://github.com/volcengine/veaiops.