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Electrical resistivity measurements were performed on single crystals of URu_2–xOs_xSi₂up tox= 0.28 under hydrostatic pressure up toP= 2 GPa. As the Os concentration,x, is increased, 1) the lattice expands, creating an effective negative chemical pressureP_ch(x); 2) the hidden-order (HO) phase is enhanced and the system is driven toward a large-moment antiferromagnetic (LMAFM) phase; and 3) less external pressureP_cis required to induce the HO→LMAFM phase transition. We compare the behavior of theT(x,P) phase boundary reported here for the URu_2-xOs_xSi₂system with previous reports of enhanced HO in URu₂Si₂upon tuning withPor similarly in URu_2–xFe_xSi₂upon tuning with positiveP_ch(x). It is noteworthy that pressure, Fe substitution, and Os substitution are the only known perturbations that enhance the HO phase and induce the first-order transition to the LMAFM phase in URu₂Si₂. We present a scenario in which the application of pressure or the isoelectronic substitution of Fe and Os ions for Ru results in an increase in the hybridization of the U-5f-electron and transition metald-electron states which leads to electronic instability in the paramagnetic phase and the concurrent formation of HO (and LMAFM) in URu₂Si₂. Calculations in the tight-binding approximation are included to determine the strength of hybridization between the U-5f-electron states and thed-electron states of Ru and its isoelectronic Fe and Os substituents in URu₂Si₂.