In other words, I have a full-time job (and full-time home life) but would still like to earn extra bits of cash leveraging my technical skills.
Does such a thing exist?
The piecemeal contract work that is time-limited and doesn't require any prior relationship usually goes to the lowest bidder. It's you versus a hundred people in other countries with 1/10th the cost of living.
Moving to a higher paying job is the fastest way to bump up your income. Freelancing can augment it, but it's a bigger commitment than what you're describing. Most clients won't be interested in you doing little bits of work as you feel like it. They want things on predictable, rapid timelines.
I don't really do it anymore, but even after transitioning to full time development (mostly backend) I used to pick up small design jobs (such as logos). Some of that was pro-bono work for or related to open source projects, but much of it was compensated at fairly reasonable rates.
I've found that design work usually wasn't needed on a tight schedule, which made it easier to schedule around a full-time job, and having a shared technical vocabulary with the clients (and being present in the technical communities they frequented) gave me an edge in landing the business and getting paid up-front. I haven't tested this proposition for a while, but I would be surprised if it doesn't still work.
Now, you may feel that you aren't a designer, but good design isn't any harder to achieve than good code. Great design, like great code, often requires a little extra something, but we're really talking about 'doing design' rather than 'becoming a designer' here. A little coding effort can go a long way in terms of procedurally generating variations on a grid in order to narrow down the search space to a few that 'look right' even though you aren't sure why.
Unless you're very quick at what you do and can find work that suits, the hourly rate can be pretty dismal. That was when it was likely less popular, and rates may have gone down. The upsides are that the posts are well written up, challenging, and have TopCoder staff that oversees the process with a Q&A forum for each contest where questions answered, requirements clarified, materials are posted, etc.
It was fun to do totally random programming tasks. Probably 80% of the time it was super-easy, particularly helping college students with their CS homework. (Don't knock it, the students are agreeable and appreciative.)
The trouble is that you'd get jobs that were just insane. For instance some German guy wanted me to log into their production Microsoft SQL server and make changes. This is something I'm just barely qualified to do, don't want to take the risk of blowing up the server, don't know if he's really got permission from his supervisor to make the change, etc.