My mantra to everyone is to just build things. Don't wait to be asked. Build things. Makers make things.
I built an ecommerce site out of curiosity using some of the lesser documented PayPal APIs and services for micro-transactions (even talking to their developers to get concrete answers.) While relaying this to someone at a tech meetup I was offered a job on the spot from a fintech company CTO because he'd been trying to solve the same problem and was impressed with how far I went to solve it out of curiosity.
Another time I got invited to beta a product from a company and was able to show them a variety of uses cases by building small projects around it, which lead to a job offer.
I wasn't doing particularly sophisticated coding, but I knew enough to build everything from mobile apps to server-sided apis.
I never would have made it past the recruiter for either company had I taken the traditional route because I've never formally worked for anyone in tech. However my habit of learning all kinds of frameworks and outside thinking was something they realized they wanted.
P.S. In the latter instance I also blogged about their technology which helped showcase its potential.