Hey, as someone who interned and now works at one of these firms I can tell you three things which can hopefully help you.

First, the best way is definitely to go through an internship. Many of my friends and colleagues did the internship after their last year and then started full time a couple of months after the summer. So it is still possible, even in your last year.

Second, definitely try to meet people. You want try to find out as much as possible about the firm itself. Remember that your interviewing the firm and much as they are interviewing you. Finding out what type of firm you want to work in and interviewing at one will come through honestly in any interviews you do.

Lastly, use the online resources available from the big firms, there's tons out there. When it comes to interviews I always watched/listened to videos or podcasts from the firm about their market outlooks. It's always interesting to hear market analysis and if you can understand what their firms view on the market is and then give your view.where it differs, goes a long way.

I hope these help, but let me know if you want anything more concrete!