Suppose I have a plug-in transformer that reduces 120VAC power to some
lower AC voltage (say 12VAC or 24VAC), and suppose I want to do the
reverse---increase a low-voltage source back up to 120VAC.

Would there be a problem in connecting the low-voltage power to the
plug-in prongs of the wall wart (instead of the normal use: plugging the
prongs into the wall outlet) and expecting to get high-voltage power
back out the other end?

I guess my question is answered if I know that there are no components
other than the transformer inside the wall wart.

My immediate need is to use a 24VCT (center-tapped, 3-wire, 12VAC or
24VAC) to convert 24VAC back up to 120VAC.