These new powers could be bequeathed upon Hercules by Zeus, Gaea or the entire Council of Godheads. All of that would make sense. The only question would be; Which new powers would Hercules get?
Aside from some form of interdimensional teleportation to get from Olympus and Earth on his own I think any new powers should be related to his fists. I do not think he should have unlimited strength that he can will to any level. That opens up a can of worms I think is best left closed. Hulk tends to be regarded either as a joke or someone who can beat anyone thanks to that potential power. His strength is already worthy of respect on a cosmic level without increasing it. Imagine how bored he would be if it increased further. Instead, Herc's powers should aid him in what he already can do. I think like Zeus he should be able to negate healing factors to some degree. It would make sense because many of the beings he fights have them on some level. I could also see him accessing his inner divine powers to hurt creatures that can normally hurt by physical attacks like the Ghost Rider. Powers like this would allow him to fight and win against opponents he might not be able to normally harm.