  We consider "new inflation" inflationary models at small fields, embedded in
minimal ${\cal N}=1$ supergravity with a single chiral superfield. Imposing a
period of inflation compatible with experiment severely restricts possible
models, classified in perturbation theory. If moreover we impose that the field
goes to large values and very small potential at the current time, like would
be needed for instance for the inflaton being the volume modulus in large extra
dimensional scenarios, the possible models are restricted to very contrived
superpotentials.
