You may also want to consider doing an internship at a publication you hope to work for full time in the future to make connections with editors and other writers at the publication. Many of these internships will likely not be paid, at least not at first. Be prepared to receive compensation in the form of connections and contacts. But be wary of being taken advantage of as free labor. If you feel uncomfortable working for free, try to find internships that pay.
It can help, but other degrees also have content writing value. For instance, the ability to synthesize and relate complex information is key to content writing, but can also be acquired from studies in education or philosophy. Companies that offer content writing positions tend to find it easier to train someone in a complex topic than to train someone to write, however. For example, as a healthcare professional with mediocre writing skills, you're less likely to be hired as a healthcare content writer than a strong writer with no healthcare background (but who can be trained in healthcare topics).