  We extend our previous work on the light-quark connected part, $a_\mu^{\rm
HVP,lqc}$, of the leading order hadronic-vacuum-polarization (HVP) contribution
to the muon anomalous magnetic moment $a_\mu$, using staggered fermions, in
several directions. We have collected more statistics on ensembles with lattice
spacings of $0.06$, $0.09$ and $0.12$ fm, and we added two new ensembles, both
with lattice spacing $0.15$ fm, but with different volumes. The increased
statistics allow us to reduce statistical errors on $a_\mu^{\rm HVP,lqc}$ and
related window quantities significantly. We also calculate the current-current
correlator from which $a_\mu^{\rm HVP,lqc}$ is obtained to
next-to-next-to-leading order (NNLO) in staggered chiral perturbation theory,
so that we can correct lattice values for $a_\mu^{\rm HVP,lqc}$ to NNLO for
finite-volume, pion-mass mistuning and taste-breaking effects. We discuss the
applicability of NNLO chiral perturbation theory to $a_\mu^{\rm HVP,lqc}$ and
to the window quantities, emphasizing that it provides a systematic EFT
approach to $a_\mu^{\rm HVP,lqc}$, but not to short- or intermediate-distance
window quantities. This makes it difficult to assess systematic errors on the
standard intermediate-distance window quantity that is now widely considered in
the literature. In view of this, we investigate a longer-distance window, for
which EFT methods should be more reliable. Our most important conclusion is
that, especially for staggered fermions, new high-statistics computations at
lattice spacings smaller than $0.06$ fm are indispensable.
