Abstract
We develop a novel general formalism allowing us to obtain values of the Hubble constant in agreement with late-time observables without degrading the fit to Cosmic Microwave Background data, considering perturbative modifications around a fiducial $\Lambda$CDM cosmology. Taking as proof-of-principle the case of a time-varying electron mass and fine structure constant, 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.