We consider a minimal extension of the Standard Model with a hidden sector charged under a dark local U(1)′ gauge group, accounting simultaneously for light neutrino masses and the observed Dark Matter relic abundance. The model contains two copies of right-handed neutrinos which give rise to light neutrino-masses via an extended seesaw mechanism. The presence of a stable Dark-Matter candidate and a massless state naturally arise by requiring the simplest anomaly-free particle content without introducing any extra symmetries. We investigate the phenomenology of the hidden sector considering the U(1)′ breaking scale of the order of the electroweak scale. Confronting the thermal history of this hidden-sector model with existing and future constraints from collider, direct and indirect detection experiments provides various possibilities of probing the model in complementary ways as every particle of the dark sector plays a specific cosmological role. Across the identified viable parameter space, a large region predicts a sizable contribution to the effective relativistic degrees-of-freedom in the early Universe that allows to alleviate the recently reported tension between late and early measurements of the Hubble constantThe work of MP was supported by the Spanish Agencia Estatal de Investigación through the grants FPA2015-65929-P (MINECO/FEDER, UE), PGC2018-095161-B-I00, IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, and Red Consolider MultiDark FPA2017-90566-REDC. MP would like to thank the Lawrence Berkeley National Laboratory for its hospitality during part of the realization of this work as well as the Paris-Saclay Particle Symposium 2019 with the support of the P2I and SPU research departments and the P2IO Laboratory of Excellence (program “Investissements d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), as well as the IPhT. This project has received funding/support from the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie grant agreements Elusives ITN No. 674896 and InvisiblesPlus RISE No. 690575. J.G. is supported by the US Department of Energy under Grant Contract DE-SC001270