You see, I am a big proponent of the "hat for a hat" mentality. And since those nasty pirates stole my hat, it is only fitting that I steal theirs in return. Maybe you could issue some vengeance on my behalf. I hear the pirates keep a a hat that used to belong to an old Captain of theirs stored in a silver chest on one of their inhabited islands. But be wary, the pirates that would have the hat are tougher than the ones that stole mine. Look for an island with tree stumps and pirate flags. Also, the orange tint to the island foliage is a hint that there is a band of tough pirates dug in there. To open the chest, you're going to need a silver key. If you need one of those, I can direct you to a nearby merchant who may have one for sale. Once you have the key, then find the island and steal their hat! Once you bring it back to me, we can talk rewards.
Turning in the quest:
Deliver the Hat
Aha! Oh the joy of sweet revenge. And on top of it all, this is a fine hat indeed! That Captain would be rolling over his grave if he knew I were wearing this hat. Thanks for getting for me! Here, take this. It should help you in your travels.
To begin the quest, talk to the Innkeeper and choose Helping the Innkeeper. He will say that he'd like to get some revenge on the pirates that stole his hat, and steal one back from them.
The Old Pirate's Hat you're looking for can only be found in Ironite Storage Chests on Battlemaster Settlements. These islands are quite far from the map's origin, but the Innkeeper can now give you directions to one. Simply ask Directions to... followed by Tough Pirates and he will direct you to the closest one.
Keep in mind that Ironite Storage Chests require a Silver Chest Key to open. If you do not have one, you may ask the Innkeeper to direct you to a Merchant that sells one.
Once you've found the Old Pirate's Hat, return to any Innkeeper to turn in the quest.