I've always found the perimeter over area close to the mark for jerrymandering but not quite good enough (because of the coastline paradox - forces picking an arbitrary segment size [1]).
My idea - I welcome counter arguments - is to calculate the district's center of mass and find the furthest point the district still contains from that center of mass. Then create a circle with that distance as the radius and find the percentage of _land_ (maybe even privately owned land?) within that circle that is in the district. My view is that the percentage should be something like 50% (maybe higher? Hard to say without actually designing districts and checking the numbers).
50% would mean that half of the habitable land remaining in the circle could be a different district but no more.