Measures initially designed to be single-trait often yield data that are compatible with both an essentially unidimensional factor-analysis (FA) solution and a correlated-factors solution. For these cases, this article proposes an approach aimed at providing information for deciding which of the two solutions is the most appropriate and useful. The procedures we propose are an FA extension of the "added-value" procedures initially proposed for subscale scores in educational testing. The basic principle is that the multiple FA solution is defensible when the factor score estimates of the primary factors are better measures of these factors than score estimates derived from a unidimensional or second-order solution. Methodologically, new results are obtained, and relations with factor indeterminacy measures and second-order FA are discussed. The procedures have been implemented in a noncommercial and widely known program for exploratory FA. The functioning of the proposal is assessed by means of a simulation study, and its usefulness is illustrated with a real-data example in the personality domain.