  Alkali-metal optically-pumped magnetometers are prone to inaccuracies arising
from the overlap of the average F = I + 1/2 and F = I - 1/2 ground-state Zeeman
resonances. We employ density-matrix simulations and experiments to investigate
how this hyperfine systematic error varies with spin polarization in a
$^{87}$Rb free-induction-decay (FID) magnetometer. At low spin polarizations,
($P \leq 0.5$), this effect causes single-frequency magnetic-field extraction
techniques to exhibit inaccuracies up to approximately 3.5 nT. Density-matrix
simulations reveal that this bias can be traced to the relative amplitude and
phase between the F = I $\pm$ 1/2 hyperfine ground-state manifolds in the FID
spin precession signal. We show that this systematic error can be mitigated
using either a double-frequency fitting model that accounts for the relative
amplitude and phase or synchronous-pulse pumping, that minimizes the F = 1
contribution to the FID signal. Theoretical simulations predict accuracies
within 0.5 nT for both techniques across a wide range of spin polarizations,
suggesting a sevenfold enhancement over single-frequency extraction methods.
Our experiments validate this, showcasing a variation in the extracted field
below 1 nT, a 3.5-fold improvement compared to single-frequency extraction
methods. Furthermore, the mitigation techniques demonstrate agreement of the
extracted magnetic field within 1.5 nT.
