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NaOsO₃ exhibits an unusual metal-insulator transition (MIT), that has been touted as Slater-type, though there are observations that do not entirely conform to this typing. Consistent with a Slater transition, the cell volume and the crystal symmetry show no change on cooling through the transition, but both a and b axes in this Pbnm orthorhombic perovskite show a clear kink and the splitting between them enlarges slightly. Since atomic positions, determined by fitting the intensity of state-of-the-art neutron diffraction data, exhibit no anomalies within the measurement uncertainty, the origin of the a-b splitting and the kink in the data have not yet been elucidated. Understanding the origin of these subtle structural changes across the transition temperature may help provide understanding about the electronic behavior.  In this study, we connect local structural changes to changes in the lattice parameters through (1) well-established structural modelling and (2) density functional theory (DFT) calculations. This approach successfully reveals a subtle, but significant change of the O-Os-O bond angle on cooling through T_MIT. A Madelung energy calculation suggests that this bond angle change signals electron localization in the t_2g orbital complex and this is verified by DFT which shows an electronic transition from itinerant to localized electronic behavior along with the bond angle shift. This detailed structural study adds an important, though heretofore overlooked, element in the microscopic picture of the metal-insulator transition in NaOsO₃.