  Measuring the size of permanent electric dipole moments (EDM) of a particle
or system provides a powerful tool to test Beyond-the-Standard-Model physics.
The diamagnetic $^{129}$Xe atom is one of the promising candidates for EDM
experiments due to its obtainable high nuclear polarization and its long
spin-coherence time in a homogeneous magnetic field. By measuring the spin
precession frequencies of polarized $^{129}$Xe and $^{3}$He, a new upper limit
on the $^{129}$Xe atomic EDM $d_\mathrm{A}(^{129}\mathrm{Xe})$ was reported in
Phys. Rev. Lett. 123, 143003 (2019). This writeup proposes a new evaluation
method based on global phase fitting (GPF) for analyzing the continuous phase
development of the $^{3}$He-$^{129}$Xe comagnetometer signal. The Cramer-Rao
Lower Bound on the $^{129}$Xe EDM for the GPF method is theoretically derived
and shows the benefit of achieving high statistical sensitivity without
bringing new systematic uncertainties. The robustness of the GPF method is
verified with Monte-Carlo studies. By optimizing the analysis parameters and
adding few more data that could not be analyzed with the former method, a
result of \[ { d_\mathrm{A} (^{129}\mathrm{Xe})=(1.1 \pm 3.6_\mathrm{(stat)}
\pm 2.0_\mathrm{(syst)})\times 10 ^{-28} e~\mathrm{cm}}, \] is obtained and is
used to derive the upper limit of $^{129}$Xe permanent EDM at 95\% C.L. \[
{|d_\text{A}(^{129}\text{Xe})| < 8.3 \times 10^{-28}~e~\mathrm{cm}}. \] This
limit is a factor of 1.7 smaller as compared to the previous result.
