Just thinking aloud here- I have read a little about this but have never designed or operated a hydro station- but I understand that in some cases there would be no real need for pumping at all- you'd just modulate the flow of water to the turbines and allow the upper reservoir level to rise while you're not running them. Some reservoirs can handle the variable level, and the downstream body of water can also handle the varying outflow, so some hydro facilities are used for "peaking" in this way. But most can't be used that way- their output can be modulated based on demand, but the excess water soon has to be spilled and its potential energy wasted when the full output isn't required.
The real problem with storage of any kind is that electricity is just too cheap to bother storing in most locales. Why would you want to build a facility capable of operating in this fashion unless there was a payback for the extra capital required? You'd need to build turbines capable of, say, 3x the average flow, and then operate them only 1/3 of the time.