  Higher-order hadronic corrections to the anomalous magnetic moment of the
muon have been evaluated including next-to-next-to-leading-order insertions of
hadronic vacuum polarization and next-to-leading-order corrections to hadronic
light-by-light scattering. This leaves a set of mixed leptonic and hadronic
corrections in the form of double-bubble topologies as the only remaining
hadronic effect at ${\mathcal O}(\alpha^4)$. Here, we estimate these
contributions by analyzing the respective cuts of the diagrams, suggesting that
the impact is limited to $\lesssim 1\times 10^{-11}$ and thus negligible at the
level of the final precision of the Fermilab $g-2$ experiment.
