  We present a new set of tools to derive systemic velocities for single-mode
RR~Lyrae stars from visual and near-infrared spectra. We derived scaling
relations and line-of-sight velocity templates using both APOGEE and {\it Gaia}
spectroscopic products combined with photometric $G$-band amplitudes. We
provide a means to estimate systemic velocities for the RR~Lyrae subclasses,
RRab and RRc. Our analysis indicates that the scaling relation between the
photometric and line-of-sight velocity amplitudes is nonlinear, with a break in
a linear relation occurring around 0.4mag in both the $V$-band and $G$-band
amplitudes. We did not observe such a break in the relation for the
first-overtone pulsators. Using stellar pulsation models, we further confirm
and examine the nonlinearity in scaling relation for the RRab subclass. We
observed little to no variation with stellar parameters (mass, metallicity, and
luminosity) in the scaling relation between the photometric and line-of-sight
velocity amplitudes for fundamental-mode pulsators. We observed an offset in
the scaling relation between the observations and stellar pulsation models,
mainly in the low-amplitude RR~Lyrae regime. This offset disappears when
different sets of convective parameters are used. Thus, the Fourier amplitudes
obtained from the photometry and line-of-sight velocity measurements can be
utilized to constrain convective parameters of stellar pulsation models. The
scaling relations and templates for APOGEE and {\it Gaia} data accurately
predict systemic velocities compared to literature values. In addition, our
tools derived from the {\it Gaia} spectra improve the precision of the derived
systemic velocities by approximately 50 percent and provide a better
description of the uncertainty distribution in comparison with previous
studies. Our newly derived tools will be used for RR~Lyrae variables observed
toward the Galactic bulge.
