You can ask the newspaper's editor to send you out on a tryout story. This is what I did to get my newspaper job. I had some clips from the school newspaper, but I was applying for a job where my competition had better clips that did not come from a school publication.
Doing a story trial basis, the editor will give you a pretty easy story that he would want in the paper. He'll give you a word count and a deadline. This will also show him what it will be like to work with you. When I did this, the story ran in the paper and they did pay me for it before I was even offered the job. But I know not all newspapers will pay for a tryout piece. I have a friend who was on tryout for a week at a really good newspaper. She already had great clips, but they wanted to see her work in the office and on deadline. She was not paid for that week, but she did get the job.
Anyway, giving applicants a tryout is nothing new in the newspaper world. Even if you hand in some other writing sample, I would suggest asking for a tryout. It is a real chance to prove yourself, and then you and the editor will both know if you are right for this. Good luck.