I was in a similar situation a few years ago. It took me some time to land a stable job and I don't have any easy solution or very original advice but I can tell you a bit about what did work and did not work for me.
Specifically, I don't think “setting your sights too high” is a problem or that seeking entry-level positions is a good strategy. It sounds like common sense and possibly dovetails with a stereotype (that of the arrogant academic expecting everything to come to them for free) but it just doesn't work in my experience.
I applied for many of these jobs, was genuinely enthusiastic about some but if you are overqualified, employers will always be concerned about the fact you could be unsatisfied with the scope of the work, the compensation package, or be eager to run away at the first occasion, etc. That's an obstacle that's very difficult to overcome, no matter what you say or do.
By contrast, I had more success in my applications with top-level employers (think Google, Microsoft…) and finally landed a great job in a R&D lab for a large European company. Having a PhD was explicitly listed as a plus on the offer and many of my colleagues came from a similar background. From there, I was in a perfect position to move to a more traditional career closer to the business and have a lot more opportunities now. So, whatever else you do, keep an eye open for this kind of positions and keep trying.
Beyond that, the things you could do are pretty obvious: network as much as you can, don't make your resume too research-y (no list of publications!), emphasise any business-relevant experience you might have (internships?), keep applying to hone your interview skills (even if that can also be frustrating and depressing) and try to find something relevant to do in the meantime (privately-funded applied research at the university?)