Nonetheless, it is possible to produce interesting photographs on those days. For example, you could try to take flightshots with slow shutterspeeds. If you manage to pan with the movement of the bird smoothly, the bird (or fast running mammal) will be reasonably sharp, whilst the background is rendered as pleasant streaks, giving the image a sense of movement.
Your best chances are with relatively slow flying birds, such as swans, geese or cranes. Smaller and faster birds are more difficult to track to begin with, and on dark days the contrast is so low that the camera will have big trouble acquiring initial focus and keeping the focus.
For larger birds such as this greylag goose, I found a shutterspeed of about 1/15th of a second is fast enough to keep the bird relatively sharp, and slow enough to produce the background streaks. Any faster, and the streaks won’t show. Any slower and you’ll have trouble panning smooth enough to keep the subject (relatively) sharp.
Greylag Goose pan blur; Canon 1D Mark III w. 500/4 IS and EF 2x; 1/15s at F8 and ISO 400; Gitzo tripod.