It depends on what sort of job you are looking for and at what type of company.
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.
