Were fats compounds that exhibited hysteresis?
Yes. Fats (as they occur in nature) are not at all homogeneous substances, which makes their melting/solidifying profiles chaotic. Furthermore, substances with elongated structures (fats, agar, gelatin) all have that pattern of melting at a higher temperature than they harden. This is because it takes time for the molecules to stack on each other to achieve the lowest energy state, which lowers the solidification temperature, but once they find that state, it takes quite a bit of energy to disrupt it. Fats that are completely unsaturated have side chains that are straight zig-zags in shape (think VVVVVVVV) (but without the serifs); since the zig-zags can stack fairly easily, those fats solidify at a fairly high temperature (above room temperature). The addition of unsaturated bonds causes bends in the zig-zags; those bends prevent the side-chains from stacking easily, thus causing the temperature to solidify them to decrease.
Breast milk has to be somewhat unsaturated, because it needs to be fully liquid at body temperature in order to exit the breast. Other fats in the body are more saturated.