  We construct data-driven solutions to the Hubble tension which are
perturbative modifications to the fiducial $\Lambda$CDM cosmology, using the
Fisher bias formalism. Taking as proof of principle the case of a time-varying
electron mass and fine structure constant, and focusing first on Planck CMB
data, we demonstrate that a modified recombination can solve the Hubble tension
and lower $S_8$ to match weak lensing measurements. Once baryonic acoustic
oscillation and uncalibrated supernovae data are included, however, it is not
possible to fully solve the tension with perturbative modifications to
recombination.
