Contribute to Open Source Software
Another underrated way of getting experience is through contributions to open source software. Not only does it integrate you into a software engineering team (allowing you to build collaboration skills) and get generally valuable experience to trade up, but you also get to work on something that you or others use!
Again, it's also not unheard of-- but certainly not common-- for strong OSS contributors to get hired by a firm after they've worked on the company's software for free. The author of Redux, Dan Abramov, is a good example.
