Four Island
Though your rival won't battle you this time, he will encourage you to check around the island to see if you can meet a familiar face. He will also alert you to the existence of Eggs, which can be received at the local Day Care if the right conditions are in place.
Six Island
If you go into the local Pokemon Center you will once again meet your rival, who has seen the entirety of the Sevii Islands. With that, he will announce he's returning to Kanto to train his team while he works on the Pokedex. Pay attention as there may be a Pokemon from the local area on his team when you meet next.
Indigo Plateau
While travelling the Sevii Islands it seems that your rival has also spent time improving his team. His coverage is better than ever, adding two very strong Pokemon found in the islands to an already well-rounded group. This team is the highest-leveled in the game too, mind. Defeat him once more and you will have completed the postgame storyline for FireRed and LeafGreen.
So in the postgame, he can be encountered three times, one of which you may battle him (and rematch him again if you want).
Your Rival (Serebii)
Hope I helped!
