Can't speak for Elk, but can say that with Red Deer (Which are a very similar species i.e. They interbreed with Elk.) the full moon during the rut is one of the best times.
Activity (Bugling, Roaring, mating.) seems to go up a gear. A good time to be out there.
Have known stags to roar almost without stopping, all night long, on a full moon.
Also the possibility of taking a shot at night in the right conditions. I'd have no hesitation in going after Elk in that time slot.

SOUTHERN MAN.
"GO HARD OR STAY HOME"