We improve the Geiger-Nuttall framework for $\alpha$ decay by incorporating neutron-proton double-shell and valence-nucleon corrections as the main structural effects through phenomenological correction terms. The role of a second-order energy-dependent term derived from the expansion of the barrier penetration factor is also examined. Based on global fits to experimental $\alpha$-decay data for even-even, odd-$A$, and odd-odd nuclei, the results show that the inclusion of proton shell effect, in addition to the conventional neutron-shell correction, leads to systematic improvements in the description of $\alpha$-decay half-lives. A modified shell criterion considering unpaired valence nucleons further enhances the agreement, particularly for odd-$A$ and odd-odd nuclei, indicating the importance of pairing blocking and single-particle effects. The fitted shell-correction coefficients further reveal that neutron shell effect provides a more robust and global contribution, while proton shell effect is more sensitive to local nuclear structure. In contrast, the second-order correction yields only a marginal improvement and provides a comparatively small additional contribution once valence-nucleon effects are properly included. These results demonstrate that the main improvements in $\alpha$-decay systematics arise from nuclear structure effects, while higher-order tunneling corrections play a relatively small role. The present formulation provides a concise and physically transparent framework for improving empirical predictions of $\alpha$-decay half-lives, with potential applications to heavy and superheavy nuclei.

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• Alpha decay
• Lifetimes & widths
• Nuclear forces
• Nuclear structure & decays