Hard to say without a picture, but would bet that the offset you describe is because both rifle scopes and Weaver side mounts have evolved a bit since your grandfather scoped the rifle. I ran into the same issue when mounting a scope on a Remington 1100 slug gun that came with a side mount attached to it.
The first pic shows two Weaver detachable side mounts, one old model (top of pic) and the other (bottom of pic) a recent purchase. Note the gap that exists between scope tube and mounting plate on the newer mount, which moves the scope tube higher and to the right when compared to the older one. I would guess that that this change was made in order to accommodate the turret housings, which in newer scopes are tend to be larger and extend outward around the entire circumference of the tube. Ultimately it was necessary to search eBay for an early top mount, and to use a scope that the mount would accommodate.
For reference, here are pics of the two mounts on a Rem 1100. Big difference for sure.