Do spec work for large companies, but, don't just redesign something for fun.

Find a company that has a problem with something that relates to your field (for example, a UX Consultant, like me) and then solve that problem with your redesign.

Explain what the problem was, how you solved it, and the value you created for the business by implementing your solution.

No one cares about your design, they care about the value you're creating for their business. Focus on that.