As a former software architect and manager of software development teams, the danger here is that so much "new skills" work needs to be re-worked. It's the re-work that gets expensive for big organizations. Worse, every big org I worked for had tons of people already working for them trying to make a break into IT - imagine the proverbial kid in the mailroom who just finished his college degree or some certification program and wants to work in IT now that he has training. Plus, we tend not to learn from doing - we learn from getting feedback on what we did. Giving that feedback means that, by definition, your "new skills" work costs the time of someone to give feedback, and then potentially re-do the work you just did. When training up an employee to stay with the company, this is a good investment. But if you aren't sticking around, why should I invest in you?

That being said, if you are in the US, check out https://givecamp.org/. It's exactly what you suggest - people doing work for small non-profits for free. I helped set up a chapter. But the trick is we always built teams of people - some senior, some junior - so the non-profit got something that was good. I'll also second another comment - learn about GitHub and FOSS.