  We propose a minimal extension of the Standard Model with the Peccei-Quinn
(PQ) scalar field and explain the relic density of the QCD axion through the
kinetic misalignment with a relatively small axion decay constant. To this
purpose, we consider a slow-roll inflation from the radial component of the PQ
field with the PQ conserving potential near the pole of its kinetic term and
investigate the post-inflationary dynamics of the PQ field for reheating. The
angular mode of the PQ field, identified with the QCD axion, receives a nonzero
velocity during inflation due to the PQ violating potential, evolving with an
approximately conserved Noether PQ charge. We determine the reheating
temperature from the perturbative decays and scattering processes of the
inflaton and obtain dark radiation from the axions produced from the inflaton
scattering at a testable level in the future Cosmic Microwave Background
experiments. We show the correlation between the reheating temperature, the
initial velocity of the axion and the axion decay constant, realizing the axion
kinetic misalignment for the correct relic density.
