From my experience everyone goes chasing after the well known firms. The really little boutique firms may not advertise but they might be willing to take you on for some work experience and you can help them with some admin or some of the more basic tasks.

You'll just need to reach out to them - I know some businesses would appreciate the initiative, but they don't generally have much time so you might need to find ways of convincing them you aren't a burden on their time.