Improved values of hyperfine coefficients related to the electronic spin-orbit and electron-nucleus spin-spin tensor interactions in the HD$^+$ molecular ion are obtained through numerical calculation of relativistic corrections at the $m\alpha^6$ order and radiative corrections at the $m\alpha^7 \ln(\alpha)$ order. The theoretical accuracy is improved by more than one order of magnitude. Some deviations with recent high-precision rovibrational spectroscopy experiments are observed, in contrast with the good agreement obtained in H$_2^+$.