I am a student with one semester left in a ASN program. During the second semester of my program I asked a lot of different nurses that I encountered if they had done an externship while they were in school. If they did, I asked them where. I got lucky and met someone who had done her externship at a hospital that was 45 minutes from my home. I immediately checked into it. She had told me the name of her nurse manager but I looked at the website and went with a higher-up person. (There wasn't a nurse recruiter) I got the interview.
I ended up getting the nurse technician job but I believe it was my cover letter that got me the interview.
My advice is to network, network, network. It's never too early to start! Even if you don't have a referral, you can do some research on the company's website and somehow bring in some info about their program/company to put in your letter. It will show that you have done your homework and that can make a good impression.
I also advise you to ALWAYS find out the name of an actual person and never just address the letter to "Nurse Recruiter". It might take some doing but usually you can find this info on their website or by asking the receptionist.
I've attached the cover letter I wrote (names removed) for an example.
BTW, I am applying for new grad residencies now, 6 months before I graduate. Don't wait!
Sample Cover Letter.pdf