A morphotropic phase boundary driven by epitaxial strain has been observed in lead-free multiferroic BiFeO3 thin films and the strain-driven phase transitions have been widely reported as iso-symmetric Cc-Cc by recent works. In this paper, it is suggested that the tetragonal-like BiFeO3 phase identified in epitaxial films on (001) LaAlO3 single crystal substrates is monoclinic M-C. This M-C phase is different from the M-A type monoclinic phase reported in BiFeO3 films grown on low mismatch substrates, such as SrTiO3. This is confirmed not only by synchrotron X-ray studies but also by piezoresponse force microscopy measurements. The polarization vectors of the tetragonal-like phase lie in the (100) plane, not the (1 (1) over bar0) plane as previously reported. A phenomenological analysis is proposed to explain the formation of M-C Phase. Such a low-symmetry M-C phase, with its linkage to M-A phase and the multiphase coexistence open an avenue for large piezoelectric response in BiFeO3 films and shed light on a complete understanding of possible polarization rotation paths and enhanced multiferroicity in BiFeO3 films mediated by epitaxial strain. This work may also aid the understanding of developing new lead-free strain-driven morphotropic phase boundary in other ferroic systems.