Jax scales well mid to late game, but starts weak. Before he gets boots, Sheen, and preferably a Phage, with some ranks of w and preferably his ult, he feels pretty **** UP, to be honest. I don't see that making him take longer to be viable within the match is a good thing. I'd much rather see some late game power sacrificed for a stronger early game, if you want to re-balance.
Perhaps you could adjust Relentless Assault. It seems nearly useless early game, with slow base attack speed and such low speed boost each swing, it rarely gets stacked and when it does, it's not terribly powerful... late game, it's a HUGE boost to his sustained damage that's up QUICKLY. What if it was just a 10% boost per swing capping at 6 stacks at every level, no scaling? It would boost his early game, taking the boost from 24% maximum to 60% maximum AS and stacking noticeably quicker, while lowering his late game, taking the boost down to 60% maximum from the current 84% maximum, lowering his sustained, and to a degree his burst dps from both auto attacks and his ult in end game.