I need help with curving artwork around a cylinder shape where the artwork goes off the sides

I'm aware of warp: Arch but I have issues with this feature looking realistic when it involves artwork going off the sides - I need the artwork to curve more and the perspective to be different on those edges whilst keeping the middle of the artwork the same. 

Example, I'm trying to edit a logo onto a pencil but I want the logo to wrap around the entire pencil (looking at the pencil straight on so can't use warp: Arch anyway), so want the top and bottom edges of the artwork to be a bit more squished horizontally as to look like its actually wrapping around and not just plonked on top.

I'm bad at explaining things but hopefully you get what I'm after! Thanks