How to transition from trading at small bank to BB?

Hello everyone. It seems like there's some very insightful people here, so I'll give this a shot.

I've been a money market trader at 2 small banks in NYC for the past 3 years. I trade deposits and FX and a bit of IRS. I have experience in making markets over reuters.

I have a strong desire to join a large/BB desk, but am just not sure how to get there. I didn't go to a target school. I will be applying to b-school in the fall, but I get the impression that an MBA is more oriented towards i-banking and not S&T.

I believe I have the skillset to be an asset on pretty much any flow desk, and would be happy to work credit/fi, energy, equities, or stay in fx/mm. I find trading to be enjoyable, and am not looking to jump simply for comp. I imagine that a bigger desk would be a much more interesting and exciting work environment.

How do people end up making this sort of transition? Networking events? I recognize that there is a real scarcity of trading roles out there now, but I believe that will change in the next couple years, so please try to consider my question from a more general perspective.

Thanks in advance

[moved from Traders Train]