In fact, the position of the statue is the first think to take note of. Go around it to look at various positions and angles. Since most statues are raised on plinths, the low angle shots can distort them. So, go away from the statue and use longer focal length lens to fill the frame. Or, put the camera up in raised arms with a timer. You may even hold the camera on your head with timer release setting.
In case, the background of the statue is messy, use a wider aperture to throw the background out of focus. If the statue is in a shady area try to avoid the use of flash. Rather, use a slightly longer shutter speed by holding your camera steady against a firm rest or use a tripod.
You can look for an interesting foreground since it can make most outdoor photographs look better. The foreground elements can provide a good compositional value.
Also, at times, wait for cloudy day or morning hours for better lighting or diffused lighting. If, you find some interesting clouds or a sunset in the background, you can concentrate on the upper part of statue for it to stand out.