  The dark sector may contain a dark photon that kinetically mixes with the
Standard Model photon, allowing dark matter to interact weakly with normal
matter. In previous work we analyzed the implications of this scenario for dark
matter capture by the Sun. Dark matter will gather in the core of the Sun and
annihilate to dark photons. These dark photons travel outwards from the center
of the Sun and may decay to produce positrons that can be detected by the Alpha
Magnetic Spectrometer (AMS-02) on the ISS. We found that the dark photon
parameter space accessible to this analysis is largely constrained by strong
limits on the spin-independent WIMP-nucleon cross section from direct detection
experiments. In this paper we build upon previous work by considering the case
where the dark sector contains two species of Dirac fermion that are nearly
degenerate in mass and couple inelastically to the dark photon. We find that
for small values of the mass splitting $\Delta \sim 100 ~\text{keV}$, the
predicted positron signal at AMS-02 remains largely unchanged from the
previously considered elastic case while constraints from direct detection are
relaxed, leaving a region of parameter space with dark matter mass $100
~\text{GeV} \lesssim m_X \lesssim 10 ~\text{TeV}$, dark photon mass $1
~\text{MeV} \lesssim m_{A'} \lesssim 100 ~\text{MeV}$, and kinetic mixing
parameter $10^{-9} \lesssim \varepsilon \lesssim 10^{-8}$ that is untouched by
supernova observations and fixed target experiments but where an inelastic dark
sector may still be discovered using existing AMS-02 data.
