Multiply Sectioned Bayesian Networks (MSBNs) extend single-agent Bayesian networks to the setting of multi-agent probabilistic reasoning. The MSBN global propagation is conducted through inter-agent message passing, coupled with intra-agent (local) message passing at local domains. Existing LJF-based MSBN inference algorithms require repeated full-scale local propagation, which may cause bottlenecks in a non-sparse network. We propose a novel method that conducts 1) delayed inter-agent message manipulation, and 2) partial local message propagation. Analysis shows that our approach significantly reduces the amount of local computation while maintaining the correctness of MSBN global propagation.