I focus on small firms. They don't have the resources or amount of work to hire a full-time (or multiple full-time) person for each role. If you can do many roles well, they don't have to source/vet a super-part-time person for each role.
> leaving me time and flexibility to work on my own project.
I think there's a bit of give-and-take there. Early on I try to look extra hard for any opportunity to show I can be flexible if they need something, and give them no reason to doubt my ability to deliver. Pretty soon trust is established, and I have all the time and flexibility I need to work on other projects alongside.
> The key is that I can keep getting the work with consistency.
My experience is my clients often don't know beforehand how long-term/consistent the work will be. But if I'm reliable and helpful, it usually turns into a long-term relationship.
> How are you finding part-time work?
HN seeking freelancer thread (https://news.ycombinator.com/user?id=whoishiring) got me a particularly nice gig, also meetups, and most of all staying in touch with people I've worked with before. For the latter two, I try to focus on quality over quantity. The handful of people who know me well, tend put me in touch with better matching gigs than the large group of superficial contacts.
Cold-emailing/cold-calling companies that look like they may be a good fit, also worked. Personally I find it a bit draining so I avoid cold-anything if I can.