  Among the models addressing the Hubble tension, those introducing a dynamical
dark component around recombination have been the most promising thus far.
Their study has highlighted that, in fact, cosmic microwave background (CMB)
and baryon acoustic oscillation (BAO) observations can allow for such
components before and near recombination.
  The new dynamical degree of freedom can be early dark energy (EDE) or early
modified gravity depending on its coupling to gravity. We study a new model,
$\mathcal{G}$EDE, featuring the cubic Galileon operator $X\Box\phi$ and test it
against the most recent Planck PR4 CMB and Cepheid calibrated Pantheon+ type Ia
Supernovae data. Thanks to the kinetic braiding effects, $\mathcal{G}$EDE gives
a better fit to the data, with a higher $H_0$, and is preferred over the
canonical EDE with a Bayes factor $\ln B=0.9$, despite introducing one more
parameter. This calls for further explorations of modified gravity near and
before last scattering. To facilitate these, we introduce a substantial
extension of the cosmological code \texttt{EFTCAMB} that allows to fully evolve
the background and linear dynamics of any covariant theory, oscillatory or not,
belonging to the Horndeski class.
