Before you start calling companies to request work, you should have at least one example of script coverage that you've written. Don't worry if you haven't done this in a professional capacity; simply ask a friend, relative or colleague to allow you to read and provide detailed coverage on her screenplay or manuscript.
Visit your local library or bookstore to check-out/buy relevant reference books. For an up-to-date listing of film production companies, which details names and phone numbers, get the Hollywood Creative Directory. For a list of publishers and literary agents, choose Writer's Market.
Call film production companies and ask to speak to the Director of Development or be transferred to the Development Department. Whether you are transferred or the receptionist asks you what your call is regarding, let them know that you are a freelance reader (or script analyst, depending on your experience and confidence level) and are calling to inquire if they are hiring any additional readers.
For publishers or literary agents, ask to speak to whoever is in charge of hiring readers. In these types of companies they rarely have special departments for that area, and the receptionist (or whoever answers the phone) will be able to answer your question. State that you are a freelance reader and would like to know if they have any job openings as reader or office assistant (since publishers and agents don't usually hire just readers).
Keep a log of your phone calls. Write down the company, phone number, date of call, and person you spoke to. Often they will say that they are not looking for anyone at the moment, but you can try back in a month or two. Taking notes is essential so you can follow up appropriately. Or they may tell you they don't use readers, and you can cross them off your list.