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Shallow nitrogen-vacancy (NV) centers in diamond are promising for nanomagnetometry, for they can be placed proximate to targets. To study the intrinsic magnetic properties, zero-field magnetometry is desirable. However, for shallow NV centers under zero field, the strain near diamond surfaces would cause level anticrossing between the spin states, leading to clock transitions whose frequencies are insensitive to magnetic signals. Furthermore, the charge noises from the surfaces would induce extra spin decoherence and hence reduce the magnetic sensitivity. Here, we demonstrate that the relatively strong hyperfine coupling (130 MHz) from a first-shell ^{13}C nuclear spin can provide an effective bias field to an NV center spin so that the clock-transition condition is broken and the charge noises are suppressed. The hyperfine bias enhances the dc magnetic sensitivity by a factor of 22 in our setup. With the charge noises suppressed by the strong hyperfine field, the ac magnetometry under zero field also reaches the limit set by decoherence due to the nuclear spin bath. In addition, the 130 MHz splitting of the NV center spin transitions allows relaxometry of magnetic noises simultaneously at two well-separated frequencies (∼2.870 ± 0.065 GHz), providing (low-resolution) spectral information of high-frequency noises under zero field. The hyperfine-bias-enhanced zero-field magnetometry can be combined with dynamical decoupling to enhance single-molecule magnetic resonance spectroscopy and to improve the frequency resolution in nanoscale magnetic resonance imaging.