By the way, the issue of picking the timing and manner for raising criticisms is important not just for contractors but for employees, trainees, interns, etc. as well. The main issue is not only that people won't give your views any credence until they know you better, or that it seems presumptuous to speak too soon after showing up, though those are both important. The main issue, in my view, is that until you have some real experience with the way things currently work in a particular operation, you're not in the best position to judge how to improve things.

This observation, and the amount of time you should wait, are both relative to the seriousness and obviousness of the issue. For example, a development group that plays with code changes in production because they have no dev/stage system is asking for a disaster. They should be advised of this pretty much ASAP, unless they have so few users in production, or those users are in sufficiently good communication with the dev team, that it isn't as big a deal. In that case, the issue is valid, but the urgency isn't the same.