You basically have it correct. You could activate Legion's Initiative's ability any time between declaring Aurelia, the Warleader as an attacker and the first end of combat step.
If you want your creatures to deal (and receive) combat damage during the first combat phase, you would want to wait until the end of combat step to do it. Otherwise, you could do it after blockers are declared; if there was a blocking situation that you didn't like the looks of. You could even do it before that, such as after declaring attackers, but that would sort of defeat the purpose of getting the extra combat.
This works because of 3 separate rules:
- Aurelia's ability triggers when she is declared as an attacker.
- Declare Attackers Step
508.1. First, the active player declares attackers.
508.1m Any abilities that trigger on attackers being declared trigger.
- Exiling Aurelia doesn't have any effect on her triggered ability after it triggers.
112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won’t affect the ability.
- When Aurelia returns to the battlefield, it is a new Aurelia who has never attacked before.
400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence.
If you waited until after the first end of combat step to do it, it would be too late; because you would then be in the second combat phase, and "the next combat" wouldn't be until next turn.