  We consider a class of well motivated supersymmetric models of F-term hybrid
inflation (FHI) which can be linked to the supersymmetric grand unification.
The predicted scalar spectral index n_s cannot be smaller than 0.97 and can
exceed unity including corrections from minimal supergravity, if the number of
e-foldings corresponding to the pivot scale k_*=0.002/Mpc is around 50. These
results are marginally consistent with the fitting of the three-year Wilkinson
microwave anisotropy probe data by the standard power-law cosmological model
with cold dark matter and a cosmological constant. However, n_s can be reduced
by applying two mechanisms: (i) The utilization of a quasi-canonical Kahler
potential with a convenient choice of a sign and (ii) the restriction of the
number of e-foldings that k_* suffered during FHI. In the case (i), we
investigate the possible reduction of n_s without generating maxima and minima
of the potential on the inflationary path. In the case (ii), the additional
e-foldings required for solving the horizon and flatness problems can be
generated by a subsequent stage of fast-roll [slow-roll] modular inflation
realized by a string modulus which does [does not] acquire effective mass
before the onset of modular inflation.
