I absolutely would encourage you to get some kind of hospital experience if you can now... CNA, PCT or even volunteer. You will absolutely stand out more than your peers and in my experience many of the "competitive positions," specifically stated that to be considered as a new graduate you must have hospital experience outside school clinicals (for example NYU), many others describe this as an "ideal," candidate... This isn't all hospitals, but again experience certainly helps.
I graduated in August 2011, and while I was very lucky to find a job quickly, many of my friends are still struggling to find jobs. Some residency programs require that you are licensed before you apply, and many other positions do too. What you can start doing is getting your "ducks in a row." Contact professors now (keep in mind some hospitals require that one letter be from a clinical instructor) and ask them for letters of reccomendation, they appreciate you asking them in advance. When you contact them I would also suggest that you provide a "goal statement, list of your relevant activities, etc." it gives them a framework to create the letter and tailor it to you. It would reccomend getting two reference letters.
The career center and my school was incredibly helpful in assisting me perfecting my resume and cover letter. You can also ask a professor or two to look it over and get their feedback.
I would highly reccomend getting the business cards of the nurse managers on the units you're currently doing clinicals on and asking them if you may contact them when you pass your boards. And unless the job post says otherwise, contact the nurse recruiters directly. You can do this before you're licensed; it shows interest and iniative. I think part of the reason I got an interview was because I called the hospital, got transferred to the nursing department, transferred to nursing recruitment and got the recruiters email off her voicemail (it was not listed on the website) in essence I practically hunted this woman down. The residency was also not listed on the website, I read their annual reports and newsletters etc. )The fact that I "did my hw, and went out of my way to contact her (and attached my resume and cover letter) indicated my strong interst from our very first contact.
Finally, I would definitely take the time to personalize your cover letter. Research the hospital, mention a recent program they iniated and why you find it attractive, (for example you were impressed with their professional certification rates in 2011). They want to know that you want to work at their hospital, that you're not just desperate for a job (but who are we kidding, as a new graduate in this economy you take what you can get before that first student loan payment is due, with a smile on your face and count your blessings).
Good luck to you on the rest of nursing school and passing your NCLEX (questions, questions, questions) and on your job hunt! You're very wise to start preparing early on and I'm sure it will serve you well in the future!!