If you're really desperate, you could try sites like oDesk or similar.
I suggest building a proposal of what you would do to improve X at Company A. Say, for example, you're interested in [role] at target companies A, B, and C. Providing them with 'free', outsider consulting is a good way to a) validate your expertise, b) show that you're interested in the company and c) start a conversation out of the blue, and perhaps make a few contacts quickly.
Try to back your way into an understanding of their strategy and tactics for X through a lot of observation and research (shouldn't actually take that long). Then, do the same for their competition, or find analogous examples in another industry. After you have a feel for their strengths and weaknesses, you could identify the gap between what their competitors do really well and what Company A does not. Then, make a thoughtful, quantitative approach for how you would improve the company's standing in X, or redesign page Y to increase R based on what you know about T. There are a lot of ways you could take this.
Summary: demonstrate your understanding of the company, the problem you want to help them solve, and your solution. Send it a person in the respective department or even Tweet a link to your proposal at the company. Best case scenario: they hire you. Worst case scenario: they don't, but other companies take notice and you're building a portfolio.
If your lease is the only thing holding you there, you might ask your landlord if you can leave early. It can depend on the landlord, and the local real estate market. Recruiting someone to take over the apartment can make the idea more amenable to a landlord as well.