cart method is the only method ^^ Except you like to do things manually
You set up a cart stockpile once, and then for all the years to come you will never have to do anything about it.
And yes, it is possible even after creating a huge mess, considering that my dwarven fortresses stockpiled EVERY item... this means including rock boulders, above ground refuse etc... There was NOTHING not stockpiled... Oh, and i stripmined about 7 zlevel, so not a single tile was unmined...^^ Add >6k bars, 10k unused blocks...
But it will probably take a while, until all items are hauled into the new QSP. But you can change the speed of the process by increasing decreasing the feeding stockpile... If you need more dwarves doing their normal work, make a small feeder stockpile (like 2x2 or 3x1), or if you have many unemployed then make a big one (like 4x4 or even 5x5)
Basically the size of the feederstockpile equals the maximum number of dwarves bringing items to the QSP.