$alpha$ decay is usually associated with both ground and low-lying isomeric states of heavy and superheavy nuclei, and the unpaired nucleon plays a key role on $alpha$ decay. In this work, we systematically studied the $alpha$ decay half-lives of odd-$A$ nuclei, including both favored and unfavored $alpha$ decay within the two-potential approach based on the isospin dependent nuclear potential. The $alpha$ preformation probabilities are estimated by using an analytic formula taking into account the shell structure and proton-neutron correlation, and the parameters are obtained through the $alpha$ decay half-lives data. The results indicate that in general the $alpha$ preformation probabilities of even-$Z$, odd-$N$ nuclei are slightly smaller than the odd-$Z$, even-$N$ ones. We found that the odd-even staggering effect may play a more important role on spontaneous fission than $alpha$ decay. The calculated half-lives can well reproduce the experimental data.
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