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 H0ΛH0H0=70km s-1Mpc-1, within the H0ΛH0H0=70km s-1Mpc-1CDM, to alleviate the trouble with H0ΛH0H0=70km s-1Mpc-1. In the presence of non-standard cosmology, however, we can fully reconcile CMB and local measurements and reach H0ΛH0H0=70km s-1Mpc-1–74 H0ΛH0H0=70km s-1Mpc-1.