The mechanism of few-layer graphene growth on the technologically-relevant cubic-SiC/Si(001) substrate is uncovered using high-resolution core-level and angle-resolved photoelectron spectroscopy, low-energy electron microscopy, and micro-spot low-energy electron diffraction. The thickness of the graphitic overlayer supported on the silicon carbide substrate and related changes in the surface structure are precisely controlled by monitoring the progress of the surface graphitization in-situ during high-temperature graphene synthesis, using a combination of microspectroscopic techniques. The experimental data reveal gradual changes in the preferential graphene lattice orientations at the initial stages of the few-layer graphene growth on SiC(001) and can act as reference data for controllable growth of single-, double-, and triple-layer graphene on silicon carbide substrates.