  The Hubble tension seems to be a crisis with $\sim5\sigma$ discrepancy
between the most recent local distance ladder measurement from type Ia
supernovae calibrated by Cepheids and the global fitting constraint from the
cosmic microwave background data. To narrow down the possible late-time
solutions to the Hubble tension, we have used in a recent study [Phys. Rev. D
105, L021301 (2022)] an improved inverse distance ladder method calibrated by
the absolute measurements of the Hubble expansion rate at high redshifts from
the cosmic chronometer data, and found no appealing evidence for new physics at
the late time beyond the $\Lambda$CDM model characterized by a parametrization
based on the cosmic age. In this paper, we further investigate the perspective
of this improved inverse distance ladder method by including the late-time
matter perturbation growth data. Independent of the dataset choices, model
parametrizations, and diagnostic quantities ($S_8$ and $S_{12}$), the new
physics at the late time beyond the $\Lambda$CDM model is strongly disfavored
so that the previous late-time no-go guide for the Hubble tension is further
strengthened.
