Well, I see two different possibilities...

1. The summoner, after summoning the Final Aeon, dies. The Final Aeon is then possessed by Yu Yevon, who works to recover Sin, and in the time it takes him to do this, there is a Calm.

2. The summoner may summon the final aeon as much as they want, and when the previous Final Aeon is killed, all of the Summoner's Aeons attack them(obviously shown in the game), and the Summoner dies this way. The Final Aeon was possesed by Yu Yevon, and so they live after the summoner dies. Yu Yevon works to recover Sin, and in this time there is a Calm.

I've always believed the latter, but I can easily see how both may be possible.