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.
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