This paper presents a partial outsourcing strategy for the vehicle routing problem with stochastic demands (VRPSD), and routing reoptimization is considered for the single-vehicle case. In the VRPSD, a vehicle may arrive at a customer's location with insufficient capacity to meet the customer demand, which results in a route failure and requires a subsequent recourse action. We propose a recourse action that utilizes outsourcing to handle unmet demand, deferring from the classical recourse action. It is formulated as a Markov decision process (MDP), and an approach based on an approximate linear programming (ALP) scheme is proposed to solve it. To effectively deal with the curse of dimensionality, several algorithmic enhancements that take advantage of the problem's structure are proposed. These include lower bounding methods based on affine functions, decomposition-based value function approximations, and constraint sampling. Our approach is compared against other approaches, and the results show that our approach generally yields high-quality solutions.