Radiative strength functions (RSFs) for the (56,57)Fe nuclei below the separation energy are obtained from the 57Fe(3He,alphagamma)56Fe and 57Fe(3He,3He'gamma)57Fe reactions, respectively. An enhancement of more than a factor of 10 over common theoretical models of the soft (E(gamma) less than or approximately equal 2 MeV) RSF for transitions in the quasicontinuum (several MeV above the yrast line) is observed. Two-step cascade intensities with soft primary transitions from the 56Fe(n,2gamma)57Fe reaction confirm the enhancement.