The issue with stone seems to be a "non-problem". Your problem as you state it is that your stone is mined at one location and used all over.
Your desire is that the stone be moved "ahead of time", supposedly because then the construction would be that much faster. The thing is, this doesn't make your construction faster, it would simply hide the cost of moving the stone to when you're doing something else.
Whether you move the stone ahead of time, or move it when you need it, the manpower required for moving x amount of stone from A to B is the same. Ideally, all your townspeople are busy all the time, so the time they spend moving the stone to a supposed "staging area" is time they're not doing some other task.
Worse, if the trip from the quarry, to the staging area, to the construction site is longer than the trip from the quarry to construction site, you're wasting manpower.
What you want to do is, when you start construction, set the building as top priority (F2 - 6 ). This will tell all available laborers to drop what they are doing and move resources to the building. Once that's done, the builders will take over and finish the job. This way you get the building done as quickly as possible, with the least waste of manpower.