Once you reach Outset, the King of Red Lions finally seems to notice that day hasn’t broken yet - in fact, time itself hasn’t changed - since you left Greatfish Island. If you remember from earlier, the pirates were going to leave at dawn. If dawn doesn’t come, they won’t leave.
Therefore, you should feel to visit with everyone. However, I want you to go to the Forest of Fairies, as you will probably go there anyhow - and you should, if you want the ability to get more Rupees!
If you remember, the path to the forest is on the far left side of the left island, on the upwards-moving path. At the top, you’ll find the bridge to be broken. You’ll also find a bit more of hill on the opposite side. Get onto the highest rock on that extra hill.
Change the wind direction to west. Face the entrance to the forest and jump off the rock, using the Deku Leaf to glide over to the entrance. Now, go into the forest.