It was touched on above, but it bears repeating: When it comes to a newly created stockpile, dwarves haul the last mined stone first. Eventually, you'll get the hang of how the hauling tasks work and you'll develop the necessary tricks. I find myself giving my miners plenty of other busy work (stone detailing, weaving/butchery/other auto-jobs) for when I need an area cleaned out. The key is that they stop mining/generating stone/generating stone hauling tasks. That old stockpile you created ages ago isn't gonna touch the newly mined stone until it's already filled with other stones that are tasked to go in the pile. Creating a new pile AFTER your miners are done with an area will tell your dwarves to fill it with the newly mined stone.
I rarely use piles for plain stone initially though, job micromanagement doesn't leave enough wiggle room to allow my initial dwarves to perform unskilled actions like hauling. Except Food and Items of course.
A trick for getting rid of stone is to build wandering Mason's/Mechanic/Craftsdwarf shops near the stone you want to clear out. No stockpiles. When queued for a job in the shop, the dwarf will usually haul a nearby stone all the way to the shop, but for subsequent jobs, they'll grab stone nearest to the shop. This is especially helpful at clearing large areas like farms/dining halls if you have a Legendary Mason/Crafter. Another thing to try, is having your mason build rock blocks. It trains the Mason up, and blocks can be put in bins. Build workshops out of blocks, build bridges out of blocks, give your fortress a nice wide smooth road. I find I can never have enough rock blocks.
Also, take an immigrant Peasant, turn off all jobs save Seige Operating, build a catapult facing a wall and set it to Fire At Will. It's slow going at first as the Operator trains up, but within a few seasons, if there's enough nearby rock, he'll make Legendary. At that point, I usually enable Item/Food Hauling on them, because they're fast.
You'll hone a system of stone removal to the point that I'm at: Having to create controlled cave-in rooms to generate more stone. I'm actually pleased to receive the "Mason Cancels construct Rock Door: Needs Rock" message.