Structured stochastic multi-armed bandits provide accelerated regret rates over the standard unstructured bandit problems. Most structured bandits, however, assume the knowledge of the structural parameter such as Lipschitz continuity, which is often not available. To cope with the latent structural parameter, we consider a transfer learning setting in which an agent must learn to transfer the structural information from the prior tasks to the next task, which is inspired by practical problems such as rate adaptation in wireless link. Specifically, we propose a novel framework to provably and accurately estimate the Lipschitz constant based on previous tasks and fully exploit it for the new task at hand. We analyze the efficiency of the proposed framework in two folds: (i) our regret bound on the new task is close to that of the oracle algorithm with the full knowledge of the Lipschitz constant under mild assumptions; and (ii) the sample complexity of our estimator matches with the information-theoretic fundamental limit. Our analysis reveals a set of useful insights on transfer learning for latent Lipschitz constants such as the fundamental challenge a learner faces. Finally, our numerical evaluations confirm our theoretical findings and show the superiority of the proposed framework compared to baselines.