Just my experience, but I freelance about 20 hrs/wk for the last 5 years. I'm a programmer and I set up a profile on a freelancing platform and basically at this point, when my hours are running low, I start responding to requests for interview from prospective employers.
I'm not particularly niche, or cheap, though probably a little under market rate in the US, but I have not had much trouble getting jobs. Don't lower your rate to match what other freelancers are advertising! I don't get every job I interview for, but I just keep interviewing until I get one.
The best jobs have been "we have something working but it has bugs and needs a new feature". If they are happy, it can turn into a few hours a week or more as there is always more to do. The code is bad and most of the skill is putting up with it and working conservatively (because there's no spec or unit tests), and not cursing at actual sentient beings.
I have interviewed with agencies, but honestly it's just another layer between you and the client and they are taking a cut and messing up communication. None of them have mentioned benefits, lol.
I'm paid hourly. In the beginning, I did a couple one-price jobs and demos, but if you are confident and clear about what you can do for the client in their current predicament you probably don't need to do that. Those gigs didn't lead to anything directly but maybe they did add to my hours-worked stats in the platform.
The key is responsiveness as I think the difficulty from the employers side is finding someone who will stick around and actually follow through. I have been overconfident and then had to drop things, it happens. At this point I'm comfortable ignoring jobs that would be too much of a reach. But in the beginning you might need to take some chances on being able to learn what you need.
The downside is I kind of long for a team where someone knows more than me sometimes. That's not bragging, I don't know much, and it's a problem with trying to level up my own skills.
I get zero benefits! And have to file my own payroll taxes! Such fun. I'm fortunate to have a partner in life with a more traditional job. I'm not sure if part time would ever give benefits though, and this work is super flexible (only one client ever has wanted regular meetings or hours), so I don't feel too dumb. In the US.