The electromagnetic coupling constant, $\alpha$, is one of the fundamental parameters of the Standard Model (SM). Its value at the Z boson mass, $\alpha(M_Z)$, is of particular interest as it enters electroweak precision tests. When running $\alpha$ from low energies up to the Z mass, five orders of magnitude in precision are lost. This makes it one of the least well determined parameters of the SM at that scale. The largest source of error comes from non-perturbative hadronic effects in the low energy region. These non-perturbative effects can be determined from ab-initio calculations in lattice QCD. At higher energies, needed to match onto QCD perturbation theory, discretization errors become large. In addition, the hadronic vacuum polarization receives logarithmically-enhanced cutoff effects which render the continuum extrapolation more difficult. To better control this extrapolation at higher energies, we test a number of improvement procedures based on lattice perturbation theory. To illustrate their effect, we present a preliminary analysis of the light quark, connected contribution to the Adler function at Euclidean $Q^2=5$ GeV$^2$. The lattice results are obtained using simulations with $2+1+1$ flavors of staggered fermions at physical values of the quark masses.