Background: Discrepancies exist between the observed abundances of argon and calcium in oxygen-neon nova ejecta and those predicted by nova models. An improved characterization of the $^{38}$K($p, \gamma$)$^{39}$Ca reaction rate over the nova temperature regime ($\sim$ 0.1 -- 0.4 GK), and thus the nuclear structure of $^{39}$Ca above the proton threshold (5770.92(63) keV), is necessary to resolve these contradictions. Purpose: The present study was performed to search for low-spin proton resonances in the $^{38}$K $+$ $p$ system, and to improve the uncertainties in energies of the known astrophysically significant proton resonances in $^{39}$Ca. Method: The level structure of $^{39}$Ca was investigated via high-resolution charged-particle spectroscopy with an Enge split-pole spectrograph using the $^{40}$Ca($^{3}$He, $\alpha$)$^{39}$Ca reaction. Differential cross sections were measured over 6 laboratory angles at 21 MeV. Distorted-wave Born approximation calculations were performed to constrain the spin-parity assignments of observed levels with special attention to those significant in determination of the $^{38}$K($p, \gamma$)$^{39}$Ca reaction rate over the nova temperature regime. Results: The resonance energies corresponding to two out of three astrophysically important states at 6154(5) and 6472.2(24) keV are measured with better precision than previous charged-particle spectroscopy measurements. A tentatively new state is discovered at 5908(3) keV. The spin-parity assignments of a few of the astrophysically important resonances are determined. Conclusions: The present $^{38}$K($p, \gamma$)$^{39}$Ca upper limit thermonuclear reaction rate at 0.1 -- 0.4 GK is higher than that determined in [Physical Review C 97 (2018) 025802] by at most a factor of 1.4 at 0.1 GK.