This seems related to

Hilbert Curves.

You asked for unique paths so I assume the start and stop of the path matters and I also assume rotation and mirrors and inversions are all "different paths".

In that case, assuming there is only one curve, then there are 81 ways to start that curve, 2 ways forwards and backwards, 4 rotations, and mirroring (which I think is the same thing as inverting the direction and a rotation).

So that'd be 81 * 2 * 4 = 648 different solutions from a single Hilbert Curve. I think one could argue these are all different views of the same solution.

Perhaps there are more space filling curves beyond just the obvious one?

Edit: I suppose zig-zagging like text on a page and spirals are also curves that would work. There must be a lot more. For example you could zig-zag for a bit and then go into a spiral for the remainder.