Normally the game does not allow you to access Diamond Steppe Island until you have the hookshot. However using some clever tricks, it is possible to get inside the island early. Obtaining the Ghost Ship Chart early is useful in order to be able to enter the Ghost Ship early, which is useful to farm rupees efficiently.
It is possible to enter Diamond Steppe early by superswimming. You will want to superswim to Diamond Steppe and get under the island before it has time to load. Kill your speed under the island. When the island loads around you, you only have to swim around to be brought up the island by the water on top. This is the fastest way to enter Diamond Steppe early, but is pretty difficult.
If you have high water waves on Diamond Steppe, there are a few spots around the island where savewarping puts you on the island early. This is very risky to go for. Once on the island, perform a wind waker dive to clip under the island and use the water on top of the island to warp up.
Get chest storage and sail to Diamond Steppe. You can enter the cave early by jumping from KoRL and gliding to the island. With chest storage, you can climb up the wall.
Using the mini tornadoes near Diamond Steppe, it is possible to gain height with the leaf and then glide onto Diamond Steppe. Once you're on the lowest side of Diamond Steppe you can wind waker dive to get under the island and from there swim to where the highest point on the island is and use the water to warp up.