  We have witnessed different values of the Hubble constant being found in the
literature in the past years. Albeit, early measurements often result in an
$H_0$ much smaller than those from late-time ones, producing a statistically
significant discrepancy, and giving rise to the so-called Hubble tension. The
trouble with the Hubble constant is often treated as a cosmological problem.
However, the Hubble constant can be a laboratory to probe cosmology and
particle physics models. In our work, we will investigate if the possibility of
explaining the $H_0$ trouble using non-thermal dark matter production aided by
phantom-like cosmology is consistent with the Cosmic Background Radiation (CMB)
and Baryon Acoustic Oscillation (BAO) data. We performed a full Monte Carlo
simulation using CMB and BAO datasets keeping the cosmological parameters
$\Omega_b h^2$, $\Omega_c h^2$, $100\theta$, $\tau_{opt}$, and $w$ as priors
and concluded that a non-thermal dark matter production aided by phantom-like
cosmology yields at most $H_0=70.5$ km s$^{-1}$Mpc$^{-1}$ which is consistent
with some late-time measurements. However, if $H_0> 72$ km s$^{-1}$ Mpc$^{-1}$
as many late-time observations indicate, an alternative solution to the Hubble
trouble is needed. Lastly, we limited the fraction of relativistic dark matter
at the matter-radiation equality to be at most 1\%.
