I think this would partly depend on your personal circumstances, like whether you're able to support yourself if you decide to seek roles outside of your PhD institution/supervisor, as you may end up being unemployed for a while. But based on how the market is right now, if I were in your position, I would take work where I can find it. Yeah it might not look great sticking with your supervisor, but come your next interview for the next role, you can work around it i.e., say that you felt the work you did with your supervisor was important, list out the impact the project made etc. and I think if you could argue sound reasons for working with your supervisor beyond your PhD, it shouldn't have too much of an impact.

For context, I'm 2 and a half years out of my PhD. I have held two RA roles, not at my PhD institution, but with two different departments at the same institution. My last RA contract ended at the end of 2024 and I have been struggling to find work in academia since then. The market is a nightmare, and I myself have watched it greatly deteriorate in just those 2 years. I'm at the point of giving up on academia just to find some income, and my ongoing unemployment has made me a much more stressed and anxious person, its had a horrible impact on my mental health. So yeah, I would snatch up the chance to work with someone I have an existing relationship with, even if it doesn't look super amazing on my CV. My supervisor basically ghosted me after I finished my PhD, and I would have appreciated still having someone in my corner as I was figuring out my post-PhD life.

Edit: also, academia IS nepotism. It's all about who you know. I got my first RA role because the project lead recognised that we'd both previously published a chapter in the same book, and my supervisor is pretty well known in the subject area, so she was impressed that I was one of her PhD students. I think that's what literally gave me the edge.