Understanding how infectious diseases spatially diffuse is critical to predict and control the epidemic prevalence. However, uncovering epidemic spatial invasion is challenging due to stochastic travel of hosts and insufficient data availability. In this study, we develop a methodology for inferring global invasion pathways on metapopulation networks with the susceptible-infected-recovered (SIR) epidemics. To solve the inference problem, we infer the invasion pathways of each invasion case as a subgraph containing a part of global invasion pathways at each epidemic arrival time. We first develop a reduction approach to decrease the sizes of the dominant invasion cases, and propose a local optimization method aiming at the SIR epidemics based on epidemic maximum diffusion (EMD) to infer spatial invasion pathways of the reduced invasion cases, and reconstruct the global invasion pathways. Compared with the previous work, we reduce the computational complexity of invasion cases, and improve the calculation of epidemic diffusion likelihood and effectiveness of the algorithm for epidemic recovery. Simulations on real and synthetic metapopulation networks verify the validity of our algorithm. Finally, an empirical example of the 2009 A (H1N1) in the USA is presented to uncover the spatial invasion pathways and identify the superinvaders.