Material flow analyses (MFAs) are powerful tools for highlighting resource efficiency opportunities in supply chains. MFAs are often represented as directed graphs, with nodes denoting processes and edges representing mass flows. However, network structure uncertainty -- uncertainty in the presence or absence of flows between nodes -- is common and can compromise flow predictions. While collection of more MFA data can reduce network structure uncertainty, an intelligent data acquisition strategy is crucial to optimize the resources (person-hours and money spent on collecting and purchasing data) invested in constructing an MFA. In this study, we apply Bayesian optimal experimental design (BOED), based on the Kullback-Leibler divergence, to efficiently target high-utility MFA data -- data that minimizes network structure uncertainty. We introduce a new method with reduced bias for estimating expected utility, demonstrating its superior accuracy over traditional approaches. We illustrate these advances with a case study on the U.S. steel sector MFA, where the expected utility of collecting specific single pieces of steel mass flow data aligns with the actual reduction in network structure uncertainty achieved by collecting said data from the United States Geological Survey and the World Steel Association. The results highlight that the optimal MFA data to collect depends on the total amount of data being gathered, making it sensitive to the scale of the data collection effort. Overall, our methods support intelligent data acquisition strategies, accelerating uncertainty reduction in MFAs and enhancing their utility for impact quantification and informed decision-making.