The carvings on Mount Rushmore face south and are best photographed at sunrise and in the morning. During the late spring and early summer, Roosevelt may be shaded part of the morning by Lincoln. In the afternoon, Washington casts his shadow over Roosevelt. Autumn and winter months bring better lighting to the front. The other problem is that it is nice to have a blue sky and possibly some interesting clouds to contrast with the rocks.
Overcast lighting is nice but the whole scene then becomes just gray.
The monument is floodlit at night. It is difficult to really get a unique picture of Mount Rushmore. Most photographs are from the viewing terrace of the closely cropped four heads or the wide-angle shot framed by the flags along the Avenue of the Flags. You might try something different like photographing a reflection of the monument in the glass panes of the gift shop. The new half-mile plank walkway of the Presidential Trail takes you up to the talus slope right below the heads where you can now get some unique views.