The dissolution of room temperature ionic liquids (RTILs) in organic solvents has been shown to enhance ion dynamics. We previously used molecular dynamics (MD) simulations to study the ionic liquid ([BMIM+][Tf2N-]) in 22 unique solvents over a wide range of concentrations. By screening over a large parameter space, we reached several conclusions: (1) ion diffusivity increases monotonically as a function of increasing ionic liquid composition, (2) pure solvent diffusivity strongly correlates with ion diffusivity, and (3) conductivity predicted by the Nernst-Einstein (NE) equation has a maximum at intermediate compositions of ionic liquid. Building off this work, we now utilize the same parameter space to study the structure of ([BMIM+][Tf2N-]) solvated in organic solvents. We explore ion correlations through a number of structural and thermodynamic properties, including liquid densities, pair correlation functions, ion pairing and ion caging lifetimes, and free energy calculations. Through these analyses, we find that some solvents are much more effective at screening ion-ion interactions than others and that these differences impact the ion dynamics in these mixtures. In general, the strong pairing of ionic liquids negatively impacts transport properties, but some solvents can robustly screen these interactions, resulting in greatly enhanced ion dynamics. These results uncover trends connecting ionic liquid structure to transport, which can help in the design of new electrolytes for energy storage devices, such as electrical double layer capacitors and batteries.