  The refined values of the magnetic dipole moments of $^{185}$Re and
$^{187}$Re nuclei are obtained. For this, we perform a combined relativistic
coupled cluster and density functional theory calculation of the shielding
constant for the ReO$_4^-$ anion. In this calculation, we explicitly include
the effect of the finite nuclear magnetization distribution in the
single-particle nuclear model using the Woods-Saxon potential for the valence
nucleon. By combining the obtained value of the shielding constant
$\sigma=4069(389)$~ppm with the available experimental nuclear magnetic
resonance data we obtain the values: $\mu(^{185}{\rm Re})=3.1567(3)(12) \mu_N,
\mu(^{187}{\rm Re})=3.1891(3)(12) \mu_N$, where the first uncertainty is the
experimental one and the second is due to theory. The refined values of
magnetic moments are in disagreement with the tabulated values, $\mu(^{185}{\rm
Re})=3.1871(3) \mu_N, \mu(^{187}{\rm Re})=3.2197(3) \mu_N$, which were obtained
using the shielding constant value calculated for the atomic cation Re$^{7+}$
rather than the molecular anion. The updated values of the nuclear magnetic
moments resolve the disagreement between theoretical predictions of the
hyperfine structure of H-like rhenium ions which were based on the tabulated
magnetic moment values and available experimental measurements. Using these
experimental data we also extract the value of the parameter of nuclear
magnetization distribution introduced in [J. Chem. Phys. \textbf{153}, 114114
(2020)], which is required to predict hyperfine structure constants for rhenium
compounds.
