  We reveal a prethermal temporal regime upon suddenly quenching to the
vicinity of a quantum phase transition in the time evolution of 1D spin chains.
The prethermal regime is analytically found to be self-similar, and its
duration is governed by the ground-state energy gap. Based on analytical
insights and numerical evidence, we show that this critically prethermal regime
universally exists independently of the location of the probe site, the
presence of weak interactions, or the initial state. Moreover, the resulting
prethermal dynamics leads to an out-of-equilibrium scaling function of the
order parameter in the vicinity of the transition.
