yup, what sillybirds said, exactly. To have the roof stick out further to the sides you need to attach (longer, stick-y-out-ier) 2x4 purlins.
Putting a larger piece of thin plywood will not work - you need at least an inch (preferably more like 2) to screw into, otherwise the roofing screws will not hold for long.
What is your existing plywood roof. If it is 5/8 or 3/4" ply and you use the mfr's filler strips, screwed on from the inside, you can probably get away with just attaching the palruf directly. But if it is thinner plywood, I would be concerned and suggest adding (or substituting) 2x4 purlins to attach the palruf and its filler strips to.
If you do end up attaching purlins (or nailers, as some call them) on top of your existing plywood roof, a) make sure they are screwed into STRUCTURE, not just plywood, and/or screw OUTWARDS so the screw heads are on the plywood inside the coop, otherwise it will not be a strong connection. And b) if it were me I would make each one in two halves, with an inch or so gap in the middle, so that any water that gets into the space between the palruf and the plywood can drain down and out rather than pooling badly and rotting thru the purlins and plywood.
Truly, the mfr's filler strips will help you get a MUCH more strongly attached and lasting roof.
Good luck,
Pat