I think someone in OP’s situation is likely to have several friends and acquaintances that are at least engineering managers at this point in their career, who in many companies (not huge ones) can have a lot of influence over hiring. If you work in startups and stuff it’s highly possible that some friend or acquaintance is a founder that is currently hiring.
The main reason I’d second the advice to use their network is that I get tons and tons of unsolicited contact from developer contracting firms and basically don’t trust any of them. The only people I have contracted with are people I knew already and trusted. Also, if I did end up paying contract developers who I didn’t trust already, I’d still probably not be willing to pay any of them exceptionally unless they were a known entity, whereas someone I trust already would be less of a financial risk since I’d have a sense of what value they’d actually add.
Anyway, I think the answer to your problem is “build your network” but I always found that advice kind of silly. The actual valuable parts of your network are people who you’ve built relationships with while working, which is more of an incidental than deliberate process in most cases. I guess maybe you could be a little intentional about it though by carefully choosing where you work and who you work with, and how you engage with others at work.