We study the uncapacitated lot-sizing problem with uncertain demand and costs. The problem is modeled as a multi-stage stochastic integer program in which the evolution of the uncertain parameters is represented by a scenario tree. To solve this problem, we propose a new extension of the stochastic dual dynamic integer programming algorithm (SDDiP). This extension aims at being more computationally efficient in the management of the expected cost-to-go functions involved in the model, in particular by reducing their number and by exploiting the current knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. The algorithm is based on a partial decomposition of the problem into a set of stochastic sub-problems, each one involving a subset of nodes forming a sub-tree of the initial scenario tree. We then introduce a cutting-plane generation procedure that iteratively strengthens the linear relaxation of these sub-problems and enables the generation of additional strengthened Benders' cut, which improves the convergence of the method. We carry out extensive computational experiments on randomly generated large-size instances. Our numerical results show that the proposed algorithm significantly outperforms the SDDiP algorithm at providing good-quality solutions within the computation time limit.