  The abundance of the QCD axion is known to be suppressed if the Hubble
parameter during inflation, $H_{\rm inf}$, is lower than the QCD scale, and if
the inflation lasts sufficiently long. We show that the tight upper bound on
the inflation scale can be significantly relaxed if the eternal old inflation
is driven by the standard-model Higgs field trapped in a false vacuum at large
field values. Specifically, $H_{\rm inf}$ can be larger than $100\,$GeV if the
false vacuum is located above the intermediate scale. We also discuss the
slow-roll inflation after the tunneling from the false vacuum to the
electroweak vacuum.
