Maybe you could implement something similar: plot inertia. After each encounter, you might be more injured, but you have the narrative benefit of the stakes being higher, so the game lets you get away with cooler stuff.
I think this idea has possibilities. It may be too simplistic, but my initial thought is a +1 to all damage rolls in the next encounter that is cumulative as long as you don't take an extended rest. So complete encounter 1 and in encounter 2, you have +1 to damage. In encounter 3, you have +2 to damage. In encounter 4, you have +3 to damage, etc. The bonus resets to 0 during an extended rest, so you have an incentive to keep going despite running low on other resources.