  The Standard Cosmological Model has experienced tremendous success at
reproducing observational data by assuming a universe dominated by a
cosmological constant and dark matter in a flat geometry. However, several
studies, based on local measurements, indicate that the universe is expanding
too fast, in disagreement with the Cosmic Microwave Background. Taking into
account combined data from CMB, Baryon Acoustic Oscillation, and type Ia
Supernovae, we show that if the mechanism behind the production of dark matter
particles has at least a small non-thermal origin, one can induce larger values
of the Hubble rate $H_0$, within the $\Lambda$CDM, to alleviate the trouble
with $H_0$. In the presence of non-standard cosmology, however, we can fully
reconcile CMB and local measurements and reach $H_0=70-74\, {\rm km s^{-1}
Mpc^{-1}}$.
