Ordered attribute grammars (OAGs for short) are a useful class of attribute grammars (AGs). For some attribute grammars, even though they are not circular, OAG circularity test reports that they are not ordered and fails to generate attribute evaluators because some approximation introduces circularities (called type 3 circularities in this paper). First we discuss that it is sometimes difficult for programmers to eliminate type 3 circularities by hand. Second, to reduce this difficulty, we propose a new AG class called OAG* that produces less type 3 circularities than OAG while preserving the positive characteristic of OAG. OAG* uses a global dependency graph GDS that provides a new approximation algorithm. We obtained good results with our experimental implementation of OAG*. It is shown that OAG* is different from the existing GAG and Eli/Liga systems. Finally, two combinations of Eli/Liga and OAG* are provided.