Can I add concrete from the old posts in the holes for the new posts and pour fast setting concrete on top?
It would be appropriate to substitute your old concrete for the gravel at the bottom of the hole (assuming you broke it small enough...), but I wouldn't mess with the concrete mix itself. 

Think about it: you'd be upending the factory mix by adding more 'gravel'. Manufacturers have thought about the mix more than we can imagine, so I'd defer to them. 

While I get that hauling more bags of concrete and more waste is a pain (maybe dribble the waste concrete out in 5# increments in your household waste, if permitted), you might take comfort in amortizing your work over the life of the posts.