You need the new tank as soon as possible. Only genetics will determine how quickly an individual fish grows. Some grow super fast, others slowly. That's for all species.
As for tank size, that's determined by fish size, amount they poop, what they eat, how messy of an eater they are, how much they like to roam/move around, etc. A little fish that poops a lot will need a bigger tank than a medium fish that only poops once a week (not that such a fish exists but you get the idea).
Puffers put out a lot of waste are are horribly messy eaters. The tank recommendation is bare minimum for the fish to survive, but bigger is always better. Plus maintaining stable parameters is easier with a higher volume.