  The $\Lambda$CDM model provides an excellent fit to the CMB data. However, a
statistically significant tension emerges when its determination of the Hubble
constant $H_0$ is compared to the local distance-redshift measurements. The
axi-Higgs model, which couples an ultralight axion to the Higgs field, offers a
specific variation of the $\Lambda$CDM model. It relaxes the $H_0$ tension as
well as explains the $^7$Li puzzle in Big-Bang nucleosynthesis, the clustering
$S_8$ tension with the weak-lensing data, and the observed isotropic cosmic
birefringence in CMB. In this paper, we demonstrate how the $H_0$ and $S_8$
tensions can be relaxed simultaneously, by correlating the axion impacts on the
early and late universe. In a benchmark scenario ($m=2 \times 10^{-30}$ eV)
selected for experimental tests soon, the analysis combining the CMB+BAO+WL+SN
data yields $H_0 = 69.9 \pm 1.5$ km/s/Mpc and $S_8 = 0.8045 \pm 0.0096$.
Combining this (excluding the SN (supernovae) part) with the local
distance-redshift measurements yields $H_0 = 72.42 \pm 0.76$ km/s/Mpc, while
$S_8$ is slightly more suppressed.
