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 and seven observations from Swift’s X-Ray Telescope collected from November 5, 2019 to April 26, 2020, as the 0.5-79 keV source flux declined from 2.8e-9 to 1.6e-12 erg/cm2/s. Over the observations, we see the source transition from a soft state with a strong disk-blackbody component with an inner disk temperature of 0.6 keV to a hard state dominated by a power-law or thermal Comptonization component. The spectral features are consistent with an accreting black hole in the soft state. For early observations, a simple two-component model does not adequately describe the data, and we explore whether reflection of the high energy component, reflection of a thermal returning radiation component, blackbody emission from the plunging region, or a combination of these provide the best explanation of these spectra.