Empty container repositioning (ECR) is crucial in handling global trade imbalances by managing the flow and storage of empty containers to effectively accommodate customer demands and returns. We formulate a Markov decision process that accounts for practical characteristics of ECR, including multiple transportation modes and lead times, and uncertainty and serial correlation in net container inflows. To determine a cost-minimising repositioning policy for real-life scenarios, we employ a stochastic approximate dynamic programming approach, integrated with statistical techniques, such as convex regression and Latin hyper cube sampling. We highlight the advantages of incorporating exogenous variables into the approximation model. A case study with historical daily data on empty container in- and out-flows demonstrates the effective control of on-hand inventory levels while optimising holding and leasing costs. Moreover, we quantify the benefits of leveraging all transportation modes, with cost reduction potentials of up to 26.05%. Finally, we evaluate the robustness of our algorithm under variations in key parameters.