We present extensive ultraviolet (UV) and optical photometric and optical spectroscopic follow-up of supernova (SN) 2021gno by the ‘Precision Observations of Infant Supernova Explosions’ (POISE) project, starting less than 2 d after the explosion. Given its intermediate luminosity, fast photometric evolution, and quick transition to the nebular phase with spectra dominated by [Ca ii] lines, SN 2021gno belongs to the small family of Calcium-rich transients. Moreover, it shows double-peaked light curves, a phenomenon shared with only four other Calcium-rich events. The projected distance from the centre of the host galaxy is not as large as other objects in this family. The initial optical light-curve peaks coincide with a very quick decline of the UV flux, indicating a fast initial cooling phase. Through hydrodynamical modelling of the bolometric light curve and line velocity evolution, we found that the observations are compatible with the explosion of a highly stripped massive star with an ejecta mass of 0.8 M_⊙ and a ⁵⁶Ni mass of 0.024 M_⊙. The initial cooling phase (first light-curve peak) is explained by the presence of an extended circumstellar material comprising ∼10⁻²M_⊙ with an extension of 1100 R_⊙⁠. We discuss if hydrogen features are present in both maximum-light and nebular spectra, and their implications in terms of the proposed progenitor scenarios for Calcium-rich transients.