  We consider a cosmological set-up, based on renormalizable superpotential
terms, in which a superheavy scale F-term hybrid inflation is followed by a
Peccei-Quinn phase transition, resolving the strong CP and mu problems of the
minimal supersymmetric standard model. We show that the field which triggers
the Peccei-Quinn phase transition can remain after inflation well above the
Peccei-Quinn scale thanks to (i) its participation in the supergravity and
logarithmic corrections during the inflationary stage and (ii) the high reheat
temperature after the same period. As a consequence, its presence influences
drastically the inflationary dynamics and the universe suffers a second period
of reheating after the Peccei-Quinn phase transition. Confronting our
inflationary predictions with the current observational data, we find that, for
about the central value of the spectral index, the grand unification scale can
be identified with its supersymmetric value for the relevant coupling constant
\kappa=0.002 and, more or less, natural values, +/-(0.01-0.1), for the
remaining parameters. On the other hand, the final reheat temeperature after
the Peccei-Quinn phase transition turns out to be low enough so as the
gravitino problem is avoided.
