  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 vs. cross section parameter space, $\{m_e,
\sigma_e\}$, the "reflected" component of the DM flux is far more energetic
than the endpoint 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 $\sigma_e$ 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.
