Modular higher-order E-unification, as described in [6], produces numerous redundant solutions in many practical cases. We present a refined algorithm for finitary theories with a finite number of non-free constants that avoids most redundant solutions, analyse it theoretically, and give first experimental results. In comparison to [6], the description of the E-unification algorithm is enriched by a definition of the translation process between first and higher-order terms and an explicit handling of new variables. This explicit handling gives deeper insight into the reasons for redundant solutions and thus provides a method for their avoidance. In order to study the efficiency of this method and the performance of the modular approach in general, some benchmark examples are presented and an interpretation of their empirical evaluation is given.