Efficient lower hybrid current drive (LHCD) is demonstrated at densities up to in diverted plasmas on the Alcator C-Mod tokamak by operating at increased plasma current—and therefore reduced Greenwald density fraction. This density exceeds the nominal “LH density limit” at reported previously, above which an anomalous loss of current drive efficiency was observed. The recovery of current drive efficiency to a level consistent with engineering scalings is correlated with a reduction in density shoulders and turbulence levels in the far scrape-off layer. Concurrently, rf wave interaction with the edge and/or scrape-off-layer plasma is reduced, as indicated by a minimal broadening of the wave frequency spectrum measured at the plasma edge. These results have important implications for sustaining steady-state tokamak operation and indicate a pathway forward for implementing efficient LHCD in a reactor.