We demonstrate the merits of an unexplored precursor, tetrasilane (Si₄H₁₀), as compared to disilane (Si₂H₆) for the growth of defect-free, epitaxial hexagonal silicon (Si). We investigate the growth kinetics of hexagonal Si shells epitaxially around defect-free wurtzite gallium phosphide (GaP) nanowires. Two temperature regimes are identified, representing two different surface reaction mechanisms for both types of precursors. Growth in the low temperature regime (415 °C-600 °C) is rate limited by interaction between the Si surface and the adsorbates, and in the high temperature regime (600 °C-735 °C) by chemisorption. The activation energy of the Si shell growth is 2.4 ±0.2 eV for Si₂H₆ and 1.5 ±0.1 eV for Si₄H₁₀ in the low temperature regime. We observe inverse tapering of the Si shells and explain this phenomenon by a basic diffusion model where the substrate acts as a particle sink. Most importantly, we show that, by using Si₄H₁₀ as a precursor instead of Si₂H₆, non-tapered Si shells can be grown with at least 50 times higher growth rate below 460 °C. The lower growth temperature may help to reduce the incorporation of impurities resulting from the growth of GaP.