We carried out a systematic study of hexagonal boron nitride/graphene (h-BN/G) heterostructure growth by introducing high incorporation of a carbon (C) source on a heated cobalt (Co) foil substrate followed by boron and nitrogen sources in a molecular beam epitaxy system. With the increase of C incorporation in Co, three distinct regions of h-BN/G heterostructures were observed from region (1) where the C saturation was not attained at the growth temperature (900 °C) and G was grown only by precipitation during the cooling process to form a 'G network' underneath the h-BN film; to region (2) where the Co substrate was just saturated by C atoms at the growth temperature and a part of G growth occurs isothermally to form G islands and another part by precipitation, resulting in a non-uniform h-BN/G film; and to region (3) where a continuous layered G structure was formed at the growth temperature and precipitated C atoms added additional G layers to the system, leading to a uniform h-BN/G film. It is also found that in all three h-BN/G heterostructure growth regions, a 3 h h-BN growth at 900 °C led to h-BN film with a thickness of 1-2 nm, regardless of the underneath G layers' thickness or morphology. Growth time and growth temperature effects have been also studied.