Map art to a tapered cylinder

I've looked everywhere for an answer to this, but I'm trying to map art onto a tapered coffee thermos. Every answer I've seen uses a straight object without curves, which, smart. That would save me a lot of hastle, but this is the bed I've made and I need to figure out how to map something to it.

The line at the bottom and the top of the image should follow the curve of the thermos. Instead the image is curved like it's a square sticker being applied to a round shape (which it is, I get that).

Is there a way to pre warp an image to correctly map top a 3D surface?