Apparently, their system will not turn up all possible (valid) flight sequences on a given day from A to B. For example, they seem to ignore routings with long layovers (especially when the first flight can connect with a shorter layover).

So it's necessary to piece together an itinerary segment by segment. Sometimes the agent will be helpful. Other times the agent will say "We can't do it that way. We can only look from origin to destination." After they fail to find anything, I just say "thanks so much for trying" and call again.

Another approach I've used is to start by asking for just the critic legs. For example, I want to go A-B-C and back, but I just ask for B-C. After they find availability, I tack on the other segments.

Anyway, be flexible and be persistent. Business class is a much better deal (especially if booked before the changes).