I don't have any inside information, nor experience with the application process, nor have I ever seen anybody here who has said to have actively applied for a job with SE, but this
what else I could POSSIBLY do prove myself to the SE folks.
is pretty easy to answer: Become active on one or more of those of the sites on the network that genuinely interest you. Either as a contributor, or just as an observer in order to get familiar with the dynamics and rules of the community.
If you have time on your hands, digging through Meta's (this site's) archives to see how the sites have developed over the years might be an interesting idea - but be warned, it's a huge archive and contains a lot of noise.
I'm very sure having a thorough basic understanding of how these sites work can't be entirely wrong; a track record of some insightful contributions is probably going to be an even bigger plus. I don't know by what criteria they do their hiring, but they have always put a lot of emphasis on wanting to hire from within the community.