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We studied the transient Galactic black hole candidate MAXI J0637-430 with data from Insight-HXMT, Swift and XMM-Newton. The broad-band X-ray observations from Insight-HXMT help us constrain the power-law component. MAXI J0637-430 is located at unusually high Galactic latitude; if it belongs to the Galactic thick disk, we suggest a most likely distance <7 kpc. Compared with other black hole transients, MAXI J0637-430 is also unusual for other reasons: a fast transition to the thermal dominant state at the start of the outburst; a low peak temperature and luminosity (we estimate them at ~ 0.7 keV and <0.1 times Eddington, respectively); a short decline timescale; a low soft-to-hard transition luminosity (<0.01 times Eddington). We argue that such properties are consistent with a small binary separation, short binary period (P ~ 2 hr), and low-mass donor star (M2 ~ 0.2 M_sun). Moreover, spectral modelling shows that a single disk-blackbody component is not a good fit to the thermal emission. Soft spectral residuals, and deviations from the standard L_disk ~ T^4 in relation, suggest the need for a second thermal component. We propose and discuss various scenarios for such component, in addition to those presented in previous studies of this source. For example, a gap in the accretion disk between a hotter inner ring near the innermost stable orbit, and a cooler outer disk. Another possibility is that the second thermal component is the thermal plasma emission from an ionized outflow.