Contacts. You might not think you have any, but you really do. If you want this enough, then put as many feelers out and follow up any leads, no matter how tenuous.
Not something I've tried as I'm lucky enough to have good contacts through my parents, but once you have secured the one placement you can play a game of Chamber-Firm ping pong. If you get a mini at a Chambers, ask their instructing solicitor if you can have a placement at their Firm. When at said firm, ask one of the Barrister's they instruct for a placement, and so on..
What part of the Country are you located?