Etching with atomic hydrogen, as a preparation step before the high-temperature growth process of graphene onto a thin 3C-SiC film grown on Si(111), greatly improves the structural quality of topmost graphene layers. Pit formation and island coalescence, which are typical of graphene growth by SiC graphitization, are quenched and accompanied by widening of the graphene domain sizes to hundreds of nanometers, and by a significant reduction in surface roughness down to a single substrate bilayer. The surface reconstructions expected for graphene and the underlying layer are shown with atomic resolution by scanning tunnelling microscopy. Spectroscopic features typical of graphene are measured by core-level photoemission and Raman spectroscopy.