The characterization and applications of topological insulators depend critically on their protected surface states, which, however, can be obscured by the presence of trivial dangling bond states. Our first-principle calculations show that this is the case for the pristine (111) surface of SnTe. Yet, the predicted surface states unfold when the dangling bond states are passivated in proper chemisorption. We further extract the anisotropic Fermi velocities, penetration lengths, and anisotropic spin textures of the unfolded (Gamma) over bar -and (M) over bar -surface states, which are consistent with the theory in Zhang et al. [Phys. Rev. B 86, 081303 (2012)]. More importantly, this chemisorption scheme provides an external control of the relative energies of different Dirac nodes, which is particularly desirable in multivalley transport.