High-precision mass measurements of $^{63}$Cu, $^{64-66}$Zn, $^{65}$Ga, $^{65-67}$Ge, $^{67}$As, $^{78,81}$Br, $^{80}$Rb, and $^{79}$Sr were performed utilizing a multireflection time-of-flight mass spectrograph combined with the gas-filled recoil ion separator GARIS-II. In the case of $^{65}$Ga, a mass uncertainty of 2.1 keV, corresponding to a relative precision of $delta m / m = 3.5times10^{-8}$, was obtained and the mass value is in excellent agreement with the 2016 Atomic Mass Evaluation. For $^{67}$Ge and $^{81}$Br, where masses were previously deduced through indirect measurements, discrepancies with literature values were found. The feasibility of using this device for mass measurements of nuclides more neutron-deficient side, which have significant impact on the $rp$-process pathway, is discussed.