My first exposure to a professional kitchen was as a stagiare, in which I worked for free. This is how I went about it: I luckily met a guy who had worked at this place, so I used him as a reference, though that really isn't necessary these days anymore, a lot of kitchens have people who come in just wanting to learn. Some restaurants even charge people now to come in and take classes! (which to me is crazy, I wouldn't do it.) Anyway, I called this restaurant, asked to speak to the chef (who's name I learned beforehand, btw,) and then told him I'd just be interested in coming in a few days a weeks to learn some stuff because I was going to culinary school. He told me to come in for an interview so I did and I stayed there for 4 months. I didn't even bring up money, I let them do it, unfortunately they couldn't pay me but I learned LOADS. This was a 3-star place in nyc and the chef totally took me under his wing because he saw how serious I was. I knew how to make terrines with foie gras, gnocchis, raviolis, cook duck breast, make duck confit, make stocks, sauces, etc before I went to school. They never threw me on the line of course, but I always assisted the saucier guy at night. That was probably my best experience in a professional kitchen ever and I didn't earn a single dime. But what I learned was priceless, and the chef took the liberty to call the school for me and give me a recommendation.
Anyways, I'm not sure what it is you are looking to learn, but as far as I know, a lot of higher end places don't pay for that kind of labor because they don't have to. Putting the name of a restaurant on your resume is enough. Not to mention what you learn by doing and watching is valuable. Personally I wouldn't want to work somewhere crappy without getting paid. It really just depends on what you want and what they can offer, I wouldn't even volunteer to work for free, let them bring it up first, they might pay you unexpectedly and that would be a fine thing as an extra bonus to what you would learn!
If I were you, to get that type of job, be aggressive. I used to pound the pavement, dropping off resumes and letters. Or I would just mail it in to places. Or even just call and ask to speak to the chef. Just explain what you want and see what happens. Its true as someone stated earlier: chefs want to see passion in workers, show that and you're in.
p.s. if you do call, call between 3-5 pm, when they're not in service, otherwise you may not hear such a pleasant voice on the other end!!