In a previous study, we developed a $1.6\mathrm{\mu m}$ continuous-wave (cw) modulation laser absorption spectrometer system for ${\mathrm{CO}}_2$ sensing and demonstrated the measurement of small fluctuations in ${\mathrm{CO}}_2$ corresponding to a precision of $4\mathrm{parts}$ per million (ppm) with a measurement interval of $32\mathrm{s}$ . In this paper, we present the process to achieve this highly specific measurement by introducing important points, which have not been shown in the previous study. Following the results of preliminary experiments, we added a function for speckle averaging on the optical antenna unit. We additionally came up with some ideas to avoid the influences of etalon effects and polarization dependence in optical components. Because of the new functions, we realized a calibration precision of $0.006\mathrm{dB}$ (rms), which corresponds to a ${\mathrm{CO}}_2$ concentration precision of less than $1\mathrm{ppm}$ for a $2\mathrm{km}$ path. We also analyzed the ${\mathrm{CO}}_2$ sensing performance after the improvements described above. The measured short time fluctuation of the differential absorption optical depth was reasonably close to that calculated using the carrier-to-noise ratio of the received signal.