Back Yard Hill Drainage Issues
I would add the new drain at the edge of the grass up to the right (not two close to the trees)  and continue it to the original drains. You can tell the original drain is working because it is mushie beyond that point this would have been the location I would have used at first but it is longer thus cost more and the previous owners may not have cared about the lawn much in the winter. A new drain should do the work of drying out the lawn.