In full tensor magnetic gradient measurement system based on superconducting quantum interference devices (SQUIDs), six first-order planar-type SQUID gradiometers mounted on the six sloping facets of a hexagonal pyramid can form the probe and determine all independent components of the magnetic gradient tensor with the help of triaxial SQUID magnetometers placed near the probe. The geometrical parameters of probe, such as the inclination angle of sloping facets and $x$ -axis orientation in the probe coordinate system o-xyz, seriously impact on the sensitivities for magnetic gradient tensor invariants. We deduced that the invariants’ sensitivities depend on the maximum among five independent elements’ sensitivities. Based on the functional relationship between each independent element’s relative sensitivity and the geometrical parameters, we designed the optimal probe geometry structure with the inclination being 58.7° and the angle of the $x$ -axis relative to the projection of the first gradiometer’s baseline on the xoy plane being ±15°, when the maximum among the relative sensitivities for the five independent elements achieved the minimum. Finally, we generalized three types of geometrical inaccuracies of the gradiometers arising in assembling the probe and evaluated the influences of each type of them on the measurement accuracy for each independent element.