Just walk in the door...
We all have to do something different to get an employers attention. One thing I thought of recently -- just walk right into the company you want and ask to speak with anyone... ANYONE. It could be a 5 minute meet/greet, an invitation to get coffee or a quick Q&A about the company and how it is to work there. Instead of a resume, give this person a letter outlining why you love their brand/services and the motivation for coming in and speaking with the staff. Get an email so you can send him/her an eCopy of the letter and see that it gets to HR and 3 other staff members!
This is definately my next move. I have been looking for work for almost a year now and most companies want you to apply online only. I think we need to get back to the old ways of walking right in the door, follow up calls do not even work anymore because companies do not even allow you to call them either. It's getting to impersonal in the world.
I'm guessing it would depend on the company and the position you were applying for. Some people get turned off if you sell yourself too hard. My suggestion would be to get to know someone(s) who works for the company that you want to get into. They might be able to alert you for internal job postings and would be able to possibly make introductions for you.