We study the jet quenching parameter $\hat q$ at the initial kinetic stages of heavy-ion collisions. Using an effective kinetic theory description of the anisotropic quark-gluon plasma, we extract its pre-equilibrium evolution to close the gap in the literature between the early glasma stage and the onset of hydrodynamics. We find that the pre-hydrodynamic evolution of $\hat q$ shows little sensitivity to the initial conditions, jet energies, and models of the transverse momentum cutoff dependence on the jet energy and temperature. This allows us to smoothly connect the large values of $\hat q$ at early times, as found in glasma simulations, with the smaller thermal values during its later hydrodynamic evolution. Similarly to the glasma case, we also observe that the jet quenching parameter is enhanced along the beam axis as compared to the transverse direction during most of the kinetic evolution. This ordering thus appears to be a distinguishing feature of the pre-hydrodynamic plasma evolution and leads to an anisotropic jet momentum broadening from the initial stages.