Recently, a new mechanism to generate a naturally small electroweak scale has been proposed. It exploits the coupling of the Higgs boson to an axionlike field and a long era in the early Universe where the axion unchains a dynamical screening of the Higgs mass. We present a new realization of this idea with the new feature that it leaves no sign of new physics at the electroweak scale, and up to a rather large scale, 109 GeV, except for two very light and weakly coupled axionlike states. One of the scalars can be a viable dark matter candidate. Such a cosmological Higgs-axion interplay could be tested with a number of experimental strategies.