We consider a class of supersymmetric inflation models, in which minimal gauged F-term hybrid inflation is coupled renormalisably tothe minimal supersymmetric standard model (MSSM), with no extraingredients; we call this class the ``minimal hybrid inflationarysupersymmetric standard model'' (MHISSM). The singlet inflaton couplesto the Higgs as well as the waterfall fields, supplying the Higgsμ-term. We show how such models can exit inflation to a vacuumcharacterised by large Higgs vevs, whose vacuum energy is controlled bysupersymmetry-breaking. The true ground state is reached after anintervening period of thermal inflation along the Higgs flat direction,which has important consequences for the cosmology of the F-terminflation scenario. The scalar spectral index is reduced, with a valueof approximately 0.976 in the case where the inflaton potential isdominated by the 1-loop radiative corrections. The reheat temperaturefollowing thermal inflation is about 109 GeV, which solves thegravitino overclosure problem. A Higgs condensate reduces the cosmicstring mass per unit length, rendering it compatible with the CosmicMicrowave Background constraints without tuning the inflaton coupling. With the minimal U(1)′ gauge symmetry in the inflation sector, where one of the waterfall fields generates aright-handed neutrino mass, we investigate the Higgs thermal inflationscenario in three popular supersymmetry-breaking schemes: AMSB,GMSB and the CMSSM, focusing on the implications for the gravitinobound. In AMSB enough gravitinos can be produced to account for theobserved dark matter abundance through decays into neutralinos.In GMSB we find an upper bound on the gravitino mass of about a TeV, while in the CMSSM the thermally generated gravitinos aresub-dominant. When Big Bang Nucleosynthesis constraints are takeninto account, the unstable gravitinos of AMSB and the CMSSM musthave a mass O(10) TeV or greater, while in GMSB we find an upperbound on the gravitino mass of O(1) TeV.