Yes, some of the Mediterranean outflow does come from great depth

In 1973 Stommel et al. predicted that the Western Mediterranean Deep Water (WMDW) contributes directly to the Mediterranean outflow through the Strait of Gibraltar (Strommel et al., 1973). Hydrographic data obtained in November 1985 showed unambiguous evidence of WMDW west of the major Gibraltar sill, thus confirming the prediction.