  We consider a class of models in which minimal gauged F-term hybrid inflation
is coupled renormalisably to the minimal supersymmetric standard model (MSSM),
with no extra ingredients; we call this class the "minimal hybrid inflationary
supersymmetric standard model" (MHISSM). The singlet inflaton supplies the
Higgs mu-term, and allows an exit from inflation to a vacuum characterised by
large Higgs vevs. The true ground state is reached after an period of thermal
inflation along the Higgs flat direction. The scalar spectral index is reduced
from the standard F-term value, to approximately 0.976 in the case where the
inflaton potential is dominated by the 1-loop corrections. The reheat
temperature following thermal inflation is about 10^9 GeV, solving the
gravitino overclosure problem. A Higgs condensate reduces the cosmic string
mass per unit length, rendering it compatible with the Cosmic Microwave
Background constraints without tuning the inflaton coupling. With the minimal
U(1)' gauge symmetry in the inflaton sector, where one of the waterfall fields
generates a right-handed neutrino mass, we investigate the Higgs thermal
inflation scenario in three popular supersymmetry-breaking schemes: AMSB, GMSB
and the CMSSM, focusing on the constraints on the gravitino mass.
