Which stat determines how fast a marksman rifle will re-center for follow-up shots? Which talents / mods would be best to increase that stat?
The stat you're looking for is Accuracy. This controls the bullet spread, or the amount that your cursor will expand upon taking a shot.
There is a talent for this too, which is Balanced which increases the speed at which you reach maximum accuracy when the weapons is shouldered. In essence this means that the cursor will narrow almost immediately, and is a fantastic talent for any Marksman rifle.
As for which things improve it, any mod with +Accuracy% will help here, which is usually optics and underbarrel mods.