Do something impressive to get noticed, such as create a really neat project online (or wherever it's relevant) or write contribute an article to a popular online blog/publication.
Introduce yourself in person and online.
Find a chain of connections between you and someone that works at the company (via facebook, or linkedin), through someone who knows you well enough to recommend you, and ask to be recommended.
Start a twitter conversation with someone at the company.
Go to a public, social, event that's attended by people from the company, and introduce yourself.
Find out who is responsible for hiring and give them your resume in person.
If you can't give a hiring manager your resume in person, do the next best thing: befriend their assistant, or any assistant at the company, or anyone at the company who can help you get your foot in the door. Many companies have referral incentives for their employees, and some employees will be willing to help regardless—if you're good.
At a previous job I had recently become friends with a woman in my apartment building who was a talented programmer but didn't have a ton of experience on paper. I could tell she was great so I recommended her to the company and fought for her, even though the company didn't think she was qualified enough. They hired her anyway, on my recommendation, and it worked out very well. I wasn't anyone's assistant or even on the development team. The point is, if you can convince someone to believe in you they can be a great asset in helping you get a job.
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