What is the period after a game's main story line ends called?

Sometimes after you finish a game main storyline the game doesn't end there, and presents other opportunities to finish old quests or perhaps unlock new areas.

Examples of this are like Fallout 3 (with DLC), which allowed you to continue playing and completing quests even after the game's main storyline had completed. In fact, it opened up new quests that were previously unavailable had the main storyline not be completed. Another example is Final Fantasy 4 for the GBA; after completion of the game and the defeat of the final boss, you were allowed to access secret areas that opened on the maps as well as take on challenge bosses for fun.

I am looking for the terminology to describe this.

"Post-completion" or "post-game" is what I commonly hear. Sometimes people will qualify it as "post-main-quest" or similar.

This is distinct from "new game plus" types of replayability that simply restarts the game allowing you to keep some stats or items or whatnot.

I would call it the epilogue.

There is not widely agreed upon terminology for the example you've given (ability to play content which you could have played before completing the game after completing it). As others have already answered terms like "Epilogue" or "Post-game" are fairly common.

There are some more specific cases of "play after completion" with more well-defined terminology though.

End-game refers to content which can only be accessed after "completion". This is most commonly used in terms of MMOs where it refers to things that you can only do after reaching max level (like Raids in World of Warcraft). Another example would be Pokémon which often has features that can only be used after defeating the Elite Four and/or the Champion.

New Game Plus refers starting a new game and importing some benefit from the completed save. Usually this is something like keeping your level in an RPG, so it's like you are continuing your progression.

The LEGO series of games normally allows "free play" after the main quests are done. You may go back and complete optional objectives that you did not complete before. In addition, there may be bonuses that are impossible to get on the first playthrough. For example, perhaps level 1 has a secret requiring a specific ability to achieve, but the LEGO character with that ability only becomes available in level 4.

I've always considered this to be called the "post game".

As another example the Pokemon series has lots of content after you see the game's credits.