I've had VERY VERY good success digging up tin ore on my chocobo in the area around the ordelle's cave's entrance chasm (not down the ramp, but at the top of the ramp and around the north end of the chasm. There were a few arrowwood logs and little worms in that mix, but predominantly ores. When I was digging in that area, I only found "nothing" one two or three digs out of two stacks of greens.