As someone who has taken these sort of jobs, yes they do. There are a few catches though.
Firstly, your experience level is going to determine a lot. As I was (and am) a student, part time jobs are expected. If you're looking at a senior position, it's going to be harder to be part time. A portfolio of some sort (apps you've made, even small things) can be mighty useful.
Secondly, look at start ups/small companies (not necessarily dev houses) who need something specific done. I had a two year gig with a company that needed an Android app written alongside their iOS and web versions that they did in house. It lasted two years because they were constantly changing their designs/content management systems/requirements, and I only worked around 12 hours per week (2 6 hour days.) This will limit your scope a little, but also opens up interesting avenues: speak to a local store about getting them an app they could use.
If you do mobile apps, most small places could use a decently priced one. If you are a APT .NET person, you could write a custom billing application. There are lots of small projects to be done, and finding one that will last a while (i.e. needs updates and modifications) is best, if possible.
Next, since you've been out of the market and are a little inexperienced, what do you have to offer? If you can work unsupervised, your options are far wider. Simply having two to three small companies you make basic things for can be interesting work, and be the kind of thing you can do without constantly being treated like you're less worthy because you're part time (although perhaps that one is just me.)
Lastly, online freelance gigs work, but in person connections are worth more IMO. That's mostly because of the connections you make: I have good working relationships with my internships and part time jobs, and that will make transitioning to full time much easier.
If I'm way too late (posting because of the edit, original was a while ago) chalk it down to "new guy syndrome". :)