We advance the state-of-the-art in automated symbolic analysis of ballot secrecy for e-voting protocols by proposing a method based on analysing three conditions that together imply ballot secrecy. Our approach has two main advantages over existing automated approaches. The first is a substantial expansion of the class of protocols and threat models that can be automatically analysed: our approach can systematically deal with (a) honest authorities present in different phases, (b) threat models in which no dishonest voters occur, and (c) protocols whose ballot secrecy depends on fresh data coming from other phases. The second advantage is that our approach can significantly improve verification efficiency, as the individual conditions are often simpler to verify. E.g., for the LEE protocol, we obtain a speedup of over two orders of magnitude. We show the scope and effectiveness of our approach using ProVerif in several case studies, including the FOO, LEE, JCJ, and Belenios protocols.