In short, have paper as 'fuel' for every administrative object, add waste paper and recycled paper.
A lot of game mechanics (contracts, laws, settlements, accounts, taxes, districts, votes, deeds, wages, rents, research, and so on) implies usage of paper writings off-screen.
To better include paper milling and have some increasing upkeep to deal with in late-game, it would be quite simple and logical to use paper (and maybe also ink/dye along) in a fuel-like component in the objects managing those game mechanics to have their status active.
For example the TownFoundation would consume like 1 paper/hour.
It means the more laws and structures you have, the more you will consume paper. Since there are several government layers now, cost will increase as the game is going on, which justify investing in more effective paper milling machine and having a good reason to mill paper in late game.
Research should really use some paper also (litteraly producing research papers and books), apart from most basic ones to unlock paper milling of course.
Getting new skill sheets from skill book could also cost one paper.
To stay on the ecology thematic, consumed paper could be output as paper waste blocks to manage.
You could either :
* store it, taking space
* burn for pollution (but giving some energy)
* recycle in a machine later in the game with water input, either giving back a bit less paper or adding carton item.
Carton could be used in some advanced recipes (composites and/or recipes using composites, in modern storages, as portable small chest?)