  We present results for the first observed outburst from the transient X-ray
binary source MAXI J0637-430. This study is based on eight observations from
the Nuclear Spectroscopic Telescope Array (NuSTAR) and six observations from
the Neil Gehrels Swift Observatory X-Ray Telescope (Swift/XRT) collected from
2019 November 19 to 2020 April 26 as the 3-79 keV source flux declined from
8.2e-10 to 1.4e-12 erg/cm^2/s. We see the source transition from a soft state
with a strong disk-blackbody component to a hard state dominated by a power-law
or thermal Comptonization component. NuSTAR provides the first reported
coverage of MAXI J0637-430 above 10 keV, and these broadband spectra show that
a two-component model does not provide an adequate description of the soft
state spectrum. As such, we test whether blackbody emission from the plunging
region could explain the excess emission. As an alternative, we test a
reflection model that includes a physical Comptonization continuum. Finally, we
also test a spectral component based on reflection of a blackbody illumination
spectrum, which can be interpreted as a simple approximation to the reflection
produced by returning disk radiation due to the bending of light by the strong
gravity of the black hole. We discuss the physical implications of each
scenario and demonstrate the value of constraining the source distance.
