Producers are from a broad range of backgrounds, and tertiary study is not always a requirement; if you were wanting to keep studying, having a good understanding of the technical aspects of creating content or digital design might be helpful. If it was more towards the TV production, understanding screen and more about performing arts broadly would be helpful.
Creating a portfolio/showreel or volunteer for roles which give you exposure to the industry and help you build connections within it. Like all of the advertising roles, it is good to start looking to make connections with other people in your chosen field, and understanding the role in more detail.
If you can, learning more of those transferable skills will help you stand out when you are applying to a place you have no previous connections.