So far, from what you have said, 2 sets of tires would be the way to go.
At least 2 sets for the rear, since, although they're not rated the best, the stock tires can be used for the front, for winter driving.
The stock rears of course should not be on a 707 HP performance car for any type of spirited driving.
I have an SRT with the identical stock tires and find them inadequate for my lesser HP car.
Many HC owners I've seen go with Track Radials, which can be used for the street or drag strip, if desired. Of course being a sticky, better traction tire the track radial don't last as long.
I plan on getting a set of Mickey Thompson ET street S/S track Radials which are street legal, but pretty good for the drag strip. But you can go for something with much better traction then those stock tires without going to a track radial, if you don't plan on requiring that level of traction. As I understand it, even going with the same 275/40 size as stock, the track radials would be a big plus for traction.
A couple of examples of popular non track radials with good performance are, the Continental Extreme Contact DW, or Michelin Pilot Super Sport.
EDIT: And the Atturo tires that "txborn58" got, I've heard are good traction tires for the money.