Is it when you enter the level from the village, when you exit the level to the village, when you find the enounters, etc?
Where do you recommend to save, basically?
The best place to save is right after you see the icon of what Pokémon the outbreak is. Then you can head over to Mass Outbreak spot.
Note that you can't save in Jubilife City as it could ruin your mass outbreak and you can't save Once you actually encounter the outbreak as it will be gone once you reload the game. So, anyplace after you know the outbreak and before you encounter it is fine.