Precise measurements of the coherent elastic neutrino-nucleus scattering may allow to test for physics beyond the Standard Model in the near future. Several new experimental setups are expected to report new and precise measurements on this process in the forthcoming years. To have robust constraints on new physics, a good knowledge of the standard physics involved in this scattering is needed. We study how different experiments can give complementary information that can be combined to have such robust constraints. We illustrate this interplay by focusing in the non-standard interactions picture in combination with a measurement of the neutron mean radius.