If you really want to be at a company you can do so much better than a resume.
A few years ago I saw an early stage startup that I knew I had to be a part of. The only problem was that it was 900 miles away and I had no connection to them. The startup was Formlabs. At the time they had 10 employees, and like most startups reduced risk by hiring people they knew. They were also situated equidistant between MIT and Harvard so there was healthy local competition. The standard approach of sending in my resume and a carefully crafted cover letter didn't work, so I decided to do something dramatic.
I had been kicking around an idea for getting a job at Apple as a mechanical engineer - I was going to design and machine an enclosure for my version of an iPhone. Then I would hunt down and send it to an appropriate engineer with a note politely asking them to give me a chance. I decided to take the same approach for Formlabs. I found a YouTube video where they showed how their prototype software worked. It was an application which takes 3D models and does a bunch of stuff to convert them into toolpaths for 3D printing. With that as reference I used a long weekend to write my own implementation with a few bonus features. I made a website showcasing it and sent it in.
I got a call the next day - they were so taken aback they thought it was a hoax and had me make a video to prove it was real. Within a week, a visit to Boston and some technical grilling I was part of the team. In the years since I have reviewed thousands of applications and no one has done anything like this. People call me crazy and say things like "who does that?" - but that's the point - no one does this.