  We propose a singlet majoron model that defines an inverse seesaw mechanism
in the $\nu$ sector. The majoron $\phi$ has a mass $m_\phi\approx 0.5$ eV and a
coupling to the $\tau$ lepton similar to the one to neutrinos. In the early
universe it is initially in thermal equilibrium, then it decouples at $T\approx
500$ GeV and contributes with just $\Delta N_{\rm eff}=0.026$ during BBN. At
$T=26$ keV (final stages of BBN) a primordial magnetic field induces resonant
$\gamma \leftrightarrow \phi$ oscillations that transfer 6% of the photon
energy into majorons, implying $\Delta N_{\rm eff}=0.55$ and a 4.7% increase in
the baryon to photon ratio. At $T\approx m_\phi$ the majoron enters in thermal
contact with the heaviest neutrino and it finally decays into $\nu \bar \nu$
pairs near recombination, setting $\Delta N_{\rm eff}=0.85$. The boost in the
expansion rate at later times may relax the Hubble tension (we obtain
$H_0=(71.4\pm 0.5)$ km/s/Mpc), while the processes $ \nu\bar \nu
\leftrightarrow \phi$ suppress the free streaming of these particles and make
the model consistent with large scale structure observations. Its lifetime and
the fact that it decays into neutrinos instead of photons lets this axion-like
majoron avoid the strong bounds that affect other axion-like particles of
similar mass and coupling to photons.
