When you're looking for work, you enter the marketplace with a certain set of skills and experiences.
As you interact with the market, you're supposed to be understanding better what sills and experiences you should have had. Then you start gaining new skills and experiences to match the market.
If your market is so small as to only consist of a few name-brand companies? Your sample size is way too small. This is a numbers game, and you need to play the numbers -- not try to mold yourself into some candidate some arbitrary set of companies would like. Quite frankly, it sounds like a fool's game.
You're a smart person. Play the game according to your rules, not theirs.
As you interact with the market, you're supposed to be understanding better what sills and experiences you should have had. Then you start gaining new skills and experiences to match the market.
If your market is so small as to only consist of a few name-brand companies? Your sample size is way too small. This is a numbers game, and you need to play the numbers -- not try to mold yourself into some candidate some arbitrary set of companies would like. Quite frankly, it sounds like a fool's game.
You're a smart person. Play the game according to your rules, not theirs.