What happened after Rita stopped resetting?

Your confusion seems to stem from misunderstanding what is looping and resetting.

The war is not what is being reset - it is the personal timelines of Rita and then Cage. Their looping is because they initially die after being contaminated with some of the aliens' blood and they are grafted the looping ability.

The aliens must have either evolved or developed the ability to loop their own timelines on death so that they can adapt and learn to overcome whatever killed them initially. So, there are actually multiple timelines being reset all over the place as aliens die and reset.

All in all - it's probably not a good idea to get much further into the physical and metaphysical ramifications, because it's apt to give even Doctor Who a nasty headache...

You kind of answered your own question: the war didn't stop. It couldn't have, since it was still happening when Cage got sent to it.

The movie doesn't tell us clearly what happened between Rita's heroics and Cage's loop. But we can assume that Rita was very heroic because of getting to relive the same battles over and over until she got them "right". Then, she lost her ability to reset, so she lost her edge - she was in situations she'd never experience before. However, she was already a brave figurehead for the war effort by that point.

The war continued, Cage got enlisted, and he got into his own reset loop. He became the new hero.