The transient Galactic black hole candidate MAXI J0637–430 went through an outburst in 2019–2020 for the very first time. This outburst was active for almost 6 months from 2019 November to 2020 May. We study the spectral properties of this source during that outburst using archival data from NICER/XTI, Swift/XRT, and NuSTAR/FPM satellites/instruments. We have analyzed the source during six epochs on which simultaneous NICER–NuSTAR and Swift/XRT–NuSTAR data were available. Using both phenomenological and physical model fitting approaches, we analyzed the spectral data in the broad 0.7–70 keV energy band. We first used a combination of disk blackbody with power-law, disk blackbody with broken power-law, and disk blackbody with power-law and bulk motion Comptonization (BMC) models. For a better understanding of the accretion picture, e.g., understanding how the accretion rates change with the changing size of the perceived Compton cloud, we used the two-component advective flow (TCAF) model with the broken power-law, and TCAF with the combined power-law, BMC models. For the last three epochs, the diskbb+power-law and TCAF models were able to spectrally fit the data for acceptable χ 2/degrees of freedom (dof). However, for the first three epochs, we needed an additional component to fit spectra for acceptable χ 2/dof. From our analysis, we reported on the possible presence of another component during these first three epochs when the source was in the high soft state. This additional component in this state is best described by the BMC phenomenon. From the TCAF model fitting, we estimated the average mass of the source as .