Micky has brought this to us :

[...]

2) I found I had a woofer and an amplifier behind the back seat. It's
not in that location in sedans because the seatbacks went down for
skis etc. and it can't mount under my rear deck because I don't have
one. The shop manual is designed around the sedan, and I have the
convertible supplement, but can't remember if has anything. It's 250
pages but only about 10 or 20 are about the body.

The entire foam ring around the speaker cone had turned almost to
dust. And surprise, the speaker/amp was unplugged. I wonder if the
previous owner did that, or if it was never plugged in.

My guess is that the foam starting to fail caused the voice coil to rub
and short which in turn caused the sound to distort. This can cause bad
sound as well as causing the amp to blow a fuse. This is probably why
the amp was disconnected. You probably need a recone and not just a new
foam edge. If you don't do it just right, it will rub again and short
the new voice coil, so it might be even better to have it done by
someone used to doing this kind of repair.

Is it worth $25 for a re-foaming kit, meant specifically for this
speaker:
http://www.simplyspeakers.com/toyota...ota.html#tab-3
or should I just glue on some cloth-backed vinyl, thinner than
uphostery, meant for clothing?

No, it is hard enough to get it set just right using the correct edge
or a whole new recone kit. Normal fabrics don't work well in
applications where radial characteristics are important.