The transient-field technique has been used to measure, with considerably improved precision, the $g$ factors of the $3/{2}^{+}$ and $5/{2}^{+}$ states in $^{125}\mathrm{Te}$ at 444 and 463 keV, respectively, relative to the $g$ factor of the first excited state in $^{126}\mathrm{Te}$. Together with shell model and weak-coupling core-excitation model calculations, the $g$-factor measurements provide insight into the orbital occupation of the odd neutron for the low-excitation states in $^{125}\mathrm{Te}$. A new $9/{2}^{+}$ level at 1029 keV, together with a firm $7/{2}^{+}$ spin assignment for the level at 1018 keV, identifies candidate states for the coupling of the ${s}_{1/2}$ neutron to the ${4}^{+}$ core excitation.