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We consider the well motivated model of the (standard) supersymmetric F-termhybrid inflation (FHI) which can be realized close to the grand unificationscale. The predicted scalar spectral index $n_s$ cannot be smaller than 0.98and can exceed unity including corrections from minimal supergravity, if thenumber of e-foldings corresponding to the pivot scale $k_*=0.002/{\rm Mpc}$ isaround 50. These results are marginally consistent with the fitting of theWilkinson microwave anisotropy probe data by the standard power-lawcosmological model with cold dark matter and a cosmological constant. However,$n_s$ can be reduced by restricting the number of e-foldings that $k_*$suffered during FHI. The additional e-foldings required for solving the horizonand flatness problems can be generated by a subsequent stage of fast-roll[slow-roll] modular inflation realized by a string modulus which does [doesnot] acquire effective mass before the onset of modular inflation.

F-Term Hybrid Inflation Followed by Modular Inflation