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.
Using the increase in speed allowed with the Swift Sail, we can jump into the center of the big octo whirlpool near Diamond Steppe. This causes KoRL to sink into the center as if he got too close to the center of the whirlpool and voids Link out. For some reason he spawns on diamond steppe after the void out. After respawning on Diamond Steppe, you can roll clip into the island using the right angled ledge near the lowest waterfall (Iron Boots help with this a lot because the water is trying to push you away), and from there swim to where the highest point on the island is and use the water to warp up.
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 roll clip into the island using the right angled ledge near the lowest waterfall (Iron Boots help with this a lot because the water is trying to push you away), and from there swim to where the highest point on the island is and use the water to warp up.
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, since water waves are mostly random. Once on the island, perform a roll clip to go under the island and use the water on top of the island to warp up.
Note: If you don't have iron boots for the roll clip, what you can do instead is item sliding forward, tapping up on the control stick to change Link's speed from negative to positive and pressing ZL. Then, you can slowly go towards the ledge you want to use for the roll clip. Because you have a lot of speed, the water won't be able to push you away.