Can I add tomato paste afterwards, instead of during cooking, for a tomato sauce?
In general you need to consider the purpose of the ingredient, and what would happen to it during the cooking that it's missed.

For example, there's no point stirring a spoonful of flour into a sauce after cooking, since the flour needs heat in order to thicken the sauce.

In the case of the tomato paste, not much will happen to that particular ingredient during cooking. It may have caramelised a little, depending on the cooking method. It would have infused more deeply into some of the other ingredients. But these are subtle things. Taste your finished product. If you feel it would benefit from an extra boost of tomato flavour, by all means stir in some tomato paste.