We propose an algorithm for schema-based determinization of finite automata on words and of stepwise hedge automata on nested words. The idea is to integrate schema-based cleaning directly into automata determinization. We prove the correctness of our new algorithm and show that it is always more efficient than standard determinization followed by schema-based cleaning. Our implementation permits to obtain a small deterministic automaton for an example of an XPath query, where standard determinization yields a huge stepwise hedge automaton for which schema-based cleaning runs out of memory.