Yes, by continuing to collaborate with your PhD advisor, you could continue to do self-funded research part-time
If an academic job doesn't materialize, but a private one does, you could arrange to continue to collaborate with your advisor on that 5th day of the week.
If what you want to do closely matches with your advisor's needs, you might even be able to negotiate a part-time post-doc position. Besides the 1-day aspect, this is very similar to what sometimes happens to recent PhD graduates during the summer after graduation. They continue to work with their advisors until they start a post-doc or some other job.
If your interests are further away, you could try to arrange to be a lab 'volunteer' and use that time to develop a grant application or collect preliminary data (as long as your advisor consents). In such scenario, you would need to talk to your advisor about the IRB process to collect the preliminary data. If there are any costs or you need equipment, you could offer to pay for them out of pocket (using the money you earned from your private sector job), and work out the details of what would happen to the equipment when/if you left.
From the advisor's perspective, you would be a 1-day-a-week, low-risk (they know you), post-doc-level research associate that provides their own equipment and funding. If you offered authorship on any resulting papers and helped with grant applications, there is very little risk to the advisor.
You could continue to do this until one of the grant applications go through and you can make enough money to quit your private sector job.
If you get a grant that allows transfers to other institutions, then your prospects of getting a full-time academic research job become much better.