Is it a good idea to self-fund a postdoc for a year?
The job market is extremely competitive, and not having had an official position will not help your CV. This is not the 70s or 80s. Academia is a very different place to what it was then. You should be aware that there is a high chance you won't be able to continue in academia, whatever you do with the next year. Would you be comfortable with running down your savings and then finding you have to change course anyway? If you want to do so, I would strongly advise talking to some tenure-track and newly-tenured professors (or equivalent) about what it is really like. I know some people now are deciding that academia in its current state isn't worth the effort.

For a strategy if you do decide to self-fund:
You need your CV to show the benefit of your extra year. However, applying for maths positions, you don't actually have an extra year, because of the length of the application cycle. To some extent that will help you - you'll roughly still be applying as a new PhD. But you should be looking to have a longer list of good papers. Since getting anything published in that timeframe is basically impossible, I would suggest getting work up on the arXiv, and finding a well-respected researcher in your field as a referee, who feels able to say that they think the paper is of the standard of X journal (or words to that effect). I don't think that's massively easy though.

In addition, I would suggest getting at least one more 'thing' on your CV beyond its current state: organising a conference, writing a grant application, or some solid teaching experience.

The relative merits of going into industry instead will depend on what your field is. For pure mathematics is unlikely you'd be able to move back from industry to academia. In statistics or some applied maths, you can still be seen as a credible researcher on the basis of work in industry. The difficulty is being able to demonstrate your personal contribution to the results. Also, many potential positions will want someone who can teach as well as do research. Demonstrating that would be difficult, but I think not impossible (eg. by supporting colleagues or by tutoring as a part-time job).