Everyone in the thread has some good idea's on using Unity. However, 1 thing that hasn't been mentioned, is the new Faction system. Think about it. "Unity" the word itself describes what it should be used for imo. Which is unifying your species.
Personally, I think you should be able to "Favor" certain factions in your empire by spending a % of your unity per month. Similar to how Suppressing it uses Influence. This way you can try to sway the Factions and populace in the direction you want. It'd be expensive though, 10% of your monthly unity gain per favored faction, but it would give a boost across the board to it.
Like-wise, instead of using Influence to reform your government, it'd instead cost a base Unity amount. This also would slow down the ascension gains as governments are reformed, etc.
Another idea is to have a new policy choice. "Forced Re Education". Basically, similar to what nations did in WWII, captured nations populace would be "Re Educated" forcefully and indoctrinated. Germany did this to French, and Polish civilians, forcing them to learn German culture, national anthem, etc.
If your policy allows for "Forced Re education", then you can pay a sum of Unity per pop, to force them to change to an ethic of your choice. This would take time, and they'd be upset the whole time its happening, but if successful they are changed to what you want, and re educated. It could be a cool combo with Chemical Bliss for certain civilizations. Chemical Bliss the pop while re educating them to your ways to keep the rebellions down.