I think the answer to your question is independent of whether you are in trading or not. What you need is a networking strategy:
Identify sectorial and functional areas within MBB that are covered by specific practices, and then devise a plan to connect with insiders at the firm. Here, it is extremely important to do it properly! By no means should you start writing to some random practice-affiliated Associates or even Project Managers. Networking works best from the very top when done properly. For example, when I networked my way into McKinsey, I did only talk to Directors and Partners who then eventually directly channeled me into the interviewing process without formal application.
This approach requires some work and diligent preparation, but in my oppinion this is the most effective method by far if you don't belong to the classical target group of fresh graduates.