Evaporation determines the low-mass reach of solar-captured dark matter because that reach is controlled by the small population of particles closest to the escape threshold. We present an orbit-space calculation of the non-thermal distribution of captured dark matter in the presence of an evaporation barrier generated by a smooth in-medium attraction sourced by the solar medium. We show that the barrier not only deepens the effective potential but also reshapes the near-threshold phase-space structure, displacing the equilibrium distribution away from weakly bound, escape-prone trajectories and towards more tightly bound core-crossing orbits, thereby suppressing evaporation and lowering the evaporation mass. Although the bulk population remains near thermal equilibrium, the near-threshold tail, as reflected in the projected velocity spectrum, acquires characteristic non-thermal structure because the barrier deforms the bound orbit space and preferentially retains particles that repeatedly traverse the hot solar core. The near-threshold tail is therefore essential for determining the low-mass reach of solar dark-matter searches in the barrier regime, and our orbit-space treatment captures the relevant physics in a controlled way.