How far in advance should I start looking for a new assignment?

At least 3 to 4 weeks before your assignment ends.

The recruiters usually won't start looking until about 3 weeks before contract end. That being said, I start asking them about 5 weeks before contract end if it is somewhere I know I don't want to stay. If that is a problem(with your recruiter) , you can work with several companies.

Probably 4 weeks or greater if you don't intend to stay in our current position.

If I don’t plan on extending where I am, I start looking 5-6 weeks Achaean of time, so my recruiter can negotiate and get me submitted and credentialed in time for the start date I want.