By the powder room under the stairs going up if you don't want to loose any space and it is viable.

To add a second staircase, you have to lose something...even if it is just open floor space. So what are you willing to sacrifice in order to fit in a second set of stairs?

The most space efficient place to add stairs is usually to stack them over/under another staircase but that would mean either losing the powder room or moving it. If you move the powder-room, you have to decide where to put it and that means sacrificing something else - maybe the kitchen pantry. Or, if you were willing to have a door leading to the basement stairs open off of the great room, you could slide the powderroom over into the current hallway space - but that would mean sacrificing that side door leading to the porch. Perhaps you could move that side door to the office instead.

In my area, the vast majority of basement steps are under the other staircase.
As mentioned above, something must be sacrificed, no matter where it goes.

Just to add to what others have posted, if the powder room is moved, consider that some building codes do not permit toilet spaces to directly open into a kitchen.
I understand the stated reasoning behind that regulation but believe that it is an unnecessary intrusion and imposed by control freaks.

Since the powder room doesn't have a window on the outside wall where it currently is anyway, I'd place it between the hall and the butler's pantry and use the adjacent closet space as well - you'd still have access from the kitchen and the foyer, which I think is plenty. I'd also be tempted to extend the study to the staircase and remove the hallway, assuming the space and door to the porch into the study - depending on how the upstairs landing works, as this would get rid of a handsome handrail feature.

Would something like this work for you?