In dynamic and resource-constrained environments, such as multi-hop wireless mesh networks, traditional routing protocols often falter by relying on predetermined paths that prove ineffective in unpredictable link conditions. Shortest Anypath routing offers a solution by adapting routing decisions based on real-time link conditions. However, the effectiveness of such routing is fundamentally dependent on the quality and reliability of the available links, and predicting these variables with certainty is challenging. This paper introduces a novel approach that leverages the Deterministic Sequencing of Exploration and Exploitation (DSEE), a multi-armed bandit algorithm, to address the need for accurate and real-time estimation of link delivery probabilities. This approach augments the reliability and resilience of the Shortest Anypath routing in the face of fluctuating link conditions. By coupling DSEE with Anypath routing, this algorithm continuously learns and ensures accurate delivery probability estimation and selects the most suitable way to efficiently route packets while maintaining a provable near-logarithmic regret bound. We also theoretically prove that our proposed scheme offers better regret scaling with respect to the network size than the previously proposed Thompson Sampling-based Opportunistic Routing (TSOR).