A simple cluster model has shown itself to be outstandingly successful in describing the half-lives for emission of heavy fragments ranging from 14C to 32Si from heavy even-even nuclei. However, its predictions for the half-lives of similar decays from odd-even nuclei are systematically below the corresponding measured values or lower limits set by experiment. In view of recent experimental results on the 223Ra → 209Pb + 14C system, we suggest that many of the exotic decays of odd-even nuclei so far observed, or searched for, may be proceeding preferentially to low-lying excited states of the daughter nucleus. We show that all the discrepancies between our model predictions and the presently available data for 14C and 24Ne emission would be consistently removed if this should prove to be the case. An opportunity to extract nuclear structure information from exotic decay measurements would thus be available.