  We introduce a new method for calculating the ${\rm O}(\alpha^3)$
hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment
from ${ab-initio}$ lattice QCD. We first derive expressions suitable for
computing the higher-order contributions either from the renormalized vacuum
polarization function $\hat\Pi(q^2)$, or directly from the lattice
vector-current correlator in Euclidean space. We then demonstrate the approach
using previously-published results for the Taylor coefficients of
$\hat\Pi(q^2)$ that were obtained on four-flavor QCD gauge-field configurations
with physical light-quark masses. We obtain $10^{10} a_\mu^{\rm HVP,HO} =
-9.3(1.3)$, in agreement with, but with a larger uncertainty than,
determinations from $e^+e^- \to {\rm hadrons}$ data plus dispersion relations.
