The policy most companies have is a BULL **** argument in court for when they are accused of ripping someone off. You can send them your plot summary (one paragraph) or maybe just the the log line (one sentence) in a letter. They might send you back a letter that they don't accept unsolicited material, but that is just CYA in court. Don't worry they won't rip you off if you have it copyrighted and on file with the WGA.
I write about 10 emails per day and I've sent out postcards as well. And frankly, I've gotten three heavies (or their assistants) read my stuff. The odds aren't good but some of them (they) DO DEFINITELY read the letters (emails).
It sounds like a "Catch 22" situation; they won't read anything they don't ask for and they can't know who you are to ask for it because they don't read submissions. HOWEVER it is a facade. A game... a legal manoeuvre. I'm sure it discourages many many good young writer. So if you are in this situation, keep plugging away. Write a freakin' 100 letters per day. If your work has merit, someone will pick it up.
That might be wrong... its not the merit of the work that counts. It is the logline. Having a killer logline is more important than a good script. You can always rewrite your script but if your logline sucks, well they just slam the door on your face and you aren't heard.
Jim Toth read The Last Dog Show - http://screenplay.biz/gelert-and-the-last-dog-show/
Ryan Abboushi read Lenin's Body - http://screenplay.biz/lenin-s-body/
Matt Martin is now reading The Tarantino Heist - screenplay.biz/the-tarantino-heist/
Ironically all three for for CAA. I guess I'm bias because they have shown an interest in my work, but they seem to be open to new ideas. I would start there. If you get a rejection letter so what? You only need one success. It doesn't matter how many rejections.
I once knew a writer with over 200,000 rejection letters. He had a barn full of them and the fire department had a fit because it was a fire hazard. That was in the old days when stamps were 10 cents. Now an email is free. :-)
Hope this helps...