Dark Matter (DM) particles (in the MeV - TeV mass scales) annihilating in the Galactic halo not only produce gamma-rays via prompt radiation, but also generate abundant e+-'s, which subsequently can emit through inverse Compton scattering, bremsstrahlung and in-flight annihilation processes (collectively called `secondary emissions'). While the prompt gamma-rays concentrate at a higher energy, the secondary emissions give rise to a broad photon spectrum ranging from hard X-rays to gamma-rays, which a number of upcoming MeV telescopes (e.g., COSI, AMEGO, E-ASTROGAM and MAST) will be able to probe. I shall discuss the sensitivity of some of these upcoming telescopes for 'weak-scale' DM particles with GeV - TeV masses, as well as for 'sub-GeV' DM with MeV - GeV masses (based on realistic models). I shall show that the MeV telescopes have the potential of probing a wide region of the DM parameter space which is currently unconstrained, precisely thanks to the significant leverage provided by their sensitivity to secondary emissions. I shall also discuss the comparisons of these MeV telescope sensitivities with the reach of other experiments.