I currently work at a grocery store full time on the weekdays and at a restaurant as a hostess on the weekends. I am a second year college student in the U.S. and I feel your pain. Although I have those jobs now, believe me, they were not easy to find. Ideally, I wanted to work in a nice retail store such as Banana Republic or Gap, however, with my lack of experience, it was impossible. However, I really put myself out there, I literally found my grocery store job because it was local. I needed to buy groceries one day and discovered the store near my house, and coincidentally, there was a sign that said,"cashier needed." I walked in and talked to the manager, and was hired on the spot. The restaurant job was similar, except that I had no idea whether they were hiring or not. I've learned that it really doesn't hurt to ask, the worst they can say is no. Now I am a hostess there.
My tips for you would be to ask everywhere, and don't be afraid to start in a lower position than you expected. Personality is truly everything, so when you introduce yourself, be yourself, but really put yourself forward. Employers really like an assertive, responsible individual who is humble and willing to learn. Also, cover letter aren't too big of a deal, depending on where you apply. Because for me, I found it so much easier to apply in person. When you talk to the hiring manager in person, most likely they will just want to see a resume. However, the will truly just want to talk to you and know the kind of person you are in real life over reading a printed sheet that lists accomplishments that may or may not truly really the type of person your are. Apply to a lot, so then you'll have a lot of options if you do get some offers. I applied to over 12 before I got these.
Best of luck! I'll be here if you need any advice.