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A generic feature of the known string inflationary models is that the same physics that makes the inflaton lighter than the Hubble scale during inflation often also makes other scalars this light.
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See, however, the following for models that seek inflation at comparatively low energies: D. H. Lyth and E. D. Stewart, “Thermal inflation and the moduli problem,” Phys. Rev. D 53
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2007
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C. P. Burgess, J. M. Cline, H. Stoica and F. Quevedo, “Inflation in realistic D-brane models,” JHEP 0409
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A. D. Linde, “Particle Physics And Inflationary Cosmology,” Chur, Switzerland: Harwood (1990) 362 p. (Contemporary concepts in physics, 5); D. H. Lyth and A. Riotto, “Particle physics models of inflation and the cosmological density perturbation,” Phys. Rept. 314
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2009
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2010
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J. P. Conlon and F. Quevedo, “Kaehler moduli inflation,” JHEP 0601
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M. Cicoli, F. G. Pedro and G. Tasinato, “Poly-instanton Inflation,” JCAP 1112
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M. Cicoli and A. Mazumdar, “Reheating for Closed String Inflation,” JCAP 1009
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