The current limit on the electron's electric dipole moment, $|d_\mathrm{e}|<8.7\times 10^{-29} e {\cdotp} {\rm cm}$ (90% confidence), was set using the molecule thorium monoxide (ThO) in the $J=1$ rotational level of its $H ^3\Delta_1$ electronic state [Science $\bf 343$, 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 $H^3\Delta_1$ state both experimentally and theoretically, including dependence on $\Omega$-doublets, rotational level, and 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 $H,J=1$.
