  By measuring the nuclear magnetic spin precession frequencies of polarized
$^{129}$Xe and $^{3}$He, a new upper limit on the $^{129}$Xe atomic electric
dipole moment (EDM) $ d_\mathrm{A} (^{129}\mathrm{Xe})$ was reported in Phys.
Rev. Lett. 123, 143003 (2019). Here, we propose 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
potential benefit of our new approach. The robustness of the GPF method is
verified with Monte-Carlo studies. By optimizing the analysis parameters and
adding data that could not be analyzed with the former method, we obtain a
result of $d_\mathrm{A} (^{129}\mathrm{Xe}) = 1.1 \pm 3.6~\mathrm{(stat)} \pm
2.0~\mathrm{(syst)} \times 10^{-28}~ e~\mathrm{cm}$ in an unblinded analysis.
For the systematic uncertainty analyses, we adopted all methods from the
aforementioned PRL publication except the comagnetometer phase drift, which can
be omitted using the GPF method. The updated null result can be interpreted as
a new upper limit of $| d_\mathrm{A} (^{129}\mathrm{Xe}) | < 8.3 \times
10^{-28}~e~\mathrm{cm}$ at the 95\% C.L.
