I have worked as a single QA in a project team so I think I understand your situation (we followed Scrum principles though). There were advises to push your team towards Agile process which is good of course; however it is often not easy to organize other people, especially if you are not their manager :) Here are several steps you could take yourself in order to optimize the QA process on the early stage:

 - Try to access all formal documentation or written requirements (it is unlikely that a product is developed without any of these). Ask your PM and your Team Lead about it. At least try to collect requirements via conversations while you are waiting for features to be deployed.
 - Try creating some documentation based on what you have collected. Why not develop some tech writing skills if you have time? They could serve you well in the future. You can also share this documentation with the team and discuss it. Might appear helpful for everyone.
 - Analyze the requirements you have collected and try to make some assumptions about potential issues (e.g., this page contains a lot of input fields -> high probability of input validation bugs; this page might be accessible to unauthorized users by direct link etc.). Share your assumptions with PM.
 - Think about optimizing the requirements. E.g., some fields or buttons may be confusing for a user; may be duplicated in another place and therefore not needed; maybe conflicting with some other elements. Share your ideas with the team and discuss.