If dark matter (DM) particles are lighter than a few MeV/c^{2} and can scatter off electrons, their interaction within the solar interior results in a considerable hardening of the spectrum of galactic dark matter received on Earth. For a large range of the mass versus cross section parameter space, {m_{e},σ_{e}}, the "reflected" component of the DM flux is far more energetic than the end point of the ambient galactic DM energy distribution, making it detectable with existing DM detectors sensitive to an energy deposition of 10-10^{3} eV. After numerically simulating the small reflected component of the DM flux, we calculate its subsequent signal due to scattering on detector electrons, deriving new constraints on σ_{e} in the MeV and sub-MeV range using existing data from the XENON10/100, LUX, PandaX-II, and XENON1T experiments, as well as making projections for future low threshold direct detection experiments.