The current limit on the electron’s electric dipole moment, $|d_e|<8.7\times{10}^{-29}$ $e\phantom{\rule{0.16em}{0ex}}\mathrm{cm}$ (90% confidence), was set using the molecule thorium monoxide (ThO) in the $J=1$ rotational level of its $H^{3}\Delta_{1}$ electronic state [J. Baron et al., Science 343, 269 (2014)]. This state in ThO is very robust against systematic errors related to magnetic fields or geometric phases, due in part to its ${\Omega}$-doublet structure. These systematics can be further suppressed by operating the experiment under conditions where the $g$-factor difference between the ${\Omega}$ doublets is minimized. We consider the $g$ factors of the ThO $H{\phantom{\rule{0.16em}{0ex}}}^{3}{{\Delta}}_{1}$ state both experimentally and theoretically, including dependence on ${\Omega}$ doublets, the rotational level, and the external electric field. The calculated and measured values are in good agreement. We find that the $g$-factor difference between ${\Omega}$ doublets is smaller in $J=2$ than in $J=1$ and reaches zero at an experimentally accessible electric field. This means that the $H,J=2$ state should be even more robust against a number of systematic errors compared to $H,J=1$.