Check out Orvidius'
Exploration Saturation heatmap over on EDAstro. It shows an (approximate) percentage of stars explored in any given subsector of space.
If you want to do "true exploration" and boldly plant your Tags where no-one has Tagged before, stay away from the red and white zones on that map. Aim for the blue and black zones, the darker the better. You'll notice the Galactic Core is mostly black. If you want to find billions of Untagged systems and fly for months without seeing another CMDR's Tag, go to the Core.
If you still have your heart set on "some kind of circumnavigation", then try and fly around
inside the red rind of the galaxy. Or, pick a distance from Sag A, and run a circle around that.
The other advantage of staying away from the extreme fringes is you don't really need a super-long jump range, or much jumponium, just to reach your destinations. A standard ship using standard jumps is sufficient, and for getting there and back again faster, there should be plenty of neutron stars at +/- 1000 LY Up and Down.