We review the dynamics of spectator scalar fields non-minimally coupled to gravity in the post-inflationary Universe, with particular emphasis on scenarios where the end of inflation is followed by a period of kination. In this context, the evolution of the Ricci scalar can lead to the spontaneous breaking of discrete or continuous symmetries through tachyonic instabilities. These Hubble-induced phase transitions may cause the amplification of field fluctuations, the formation of transient topological defects, and an efficient energy transfer into relativistic degrees of freedom. We analyze this general mechanism and its cosmological consequences, including (re)heating and gravitational wave production. As a concrete realization, we discuss the postinflationary evolution of the Standard Model Higgs, examining the interplay between curvature effects, vacuum stability, and non-perturbative dynamics. This framework provides a minimal and predictive connection between high-energy physics and the early Universe, with potential observational signatures.