(presuming that you are planning to hold the material axis vertical)
The giffin grip is pretty much a soft grip, lowish precision three jaw scroll chuck. A neat implementation.
Any three scroll chuck is going to be ok, but not great, at precise centering, and you still need to address angular alignment unless the pieces are rotationally symmetric.
The workholding here has so many options, many of which depend on the particular geometry, that it is tough to give a simple answer.
For repetative, or nearly so, work, I would make a fixture to hold the part that can be mounted to the wasteboard or to the vise alignment holes in the bedplate. Dowel pins or shoulder screws for alignment will get you within 0.01 or 0.02mm on each position. The key things is machine it on the nomad in position.
Two hard points and a screw will provide location for the part. If the work needs to flip instead of the fixture, if the fixture is machined in place alignment should be fine.
If the work has holes that can be used for alignment, that is a real good way to go. Hole in the wasteboard and a dowel pin for alignment. Again, the position will be precisely known, as it will be positioned by the machine itself.
If there is no easy way to hold the work after flip, machine a pocket (a negative of the top of the work) to seat it into for the bottom side machining. Again, this gives precise positioning with reasonable ease.
You can mount jaws to the wasteboard or bedplate several ways. Mount two bars to the bed with a small gap between them at the center of the bed. Machine the profile of the work to span them, loosen one to mount the work. If the profile is machined into the bars a little bit small, when the bars are tightened down, they will grip the work.