The diffusion of small gases to special binding sites within polypeptide matrices pivotally defines the biochemical specificity and reactivity of proteins. We investigate here explicit O-2 diffusion in adult human hemoglobin (HbA) as a case study employing the recently developed temperature-controlled locally enhanced sampling (TLES) method and vary the parameters to greatly increase the simulation efficiency. The method is carefully validated against standard molecular dynamics (MD) simulations and available experimental structural and kinetic data on ligand diffusion in T-state deoxyHbA. The methodology provides a viable alternative approach to traditional MD simulations and/or potential of mean force calculations for: (i) characterizing kinetically accessible diffusion tunnels and escape routes for light ligands in porous proteins; (ii) very large systems when realistic simulations require the inclusion of multiple subunits of a protein; and (iii) proteins that access short-lived conformations relative to the simulation time. In the case of T-state deoxyHbA, we find distinct ligand diffusion tunnels consistent with the experimentally observed disparate Xe cavities in the alpha- and beta-subunits. We identify two distal barriers including the distal histidine (E7) that control access to the heme. The multiple escape routes uncovered by our simulations call for a review of the current popular hypothesis on ligand escape from hemoglobin. Larger deviations from the crystal structure during simulated diffusion in isolated alpha- and beta-subunits highlight the dampening effects of subunit interactions and the importance of including all subunits of multisubunit proteins to map realistic kinetically accessible diffusion tunnels and escape routes.