I got a sheet of thin fabric backed black vinyl, and used vinyl repair glue, and recovered the whole seat.
I also used fabric spray glue on the main area, because it covers better, but the vinyl repair glue holds much better at the perimeter where it has to grip the seat base.
If you trim away the old vinyl around where it glues to the underside, it will be easier to affix the new layer.
It isn't exactly an easy job, but a thinner material makes it easier.