Jump to Beedle's ship, pull out the Wind Waker, and perform a Wind Waker Dive. Wait a second, and then exit out of the conducting menu. Swim over under the entrance to the Deku Tree's chamber and wait to surface. When you do, you should surface above ground and in the loading zone.
A roll clip can be performed at the bottom of Forest Haven in order to clip under the island and swim to the loading zone. However, this trick is somewhat precise and also requires Beedle's ship to move slightly, making it only a small timesaver in a Wind Waker HD speedrun.
Since Wind Waker diving only works in very few places in Wind Waker HD, one alternative to a Wind Waker dive in a Hyoi Pear dive. However, since this requires the purchase of a Bait Bag, this technique is not faster in a speedrun.
By voiding-out with storage right as the deku tree cutscene loads, it is possible to cancel it and get the leaf right away, skipping the cutscene. First, get storage and roll into the deku tree. Because of the effect of storage on the cutscene of the chus dropping from the tree, the chus will be bugged and will slide accross the room. Then you will want to push the chus out of bounds so that they continue sliding OoB and fall OoB, which will kill them. By doing that, you should have enough time to void-out in the room. You can either use the deku tree's chin to get crushed (easiest way), or you can roll-clip OoB and fall into the void. Once you've voided-out, you only have to maneuver to the deku leaf and you will have successfully skipped the cutscene. Perform this first and then opt for any leaf early method if you don't have the grappling hook.
See the Early Deku Leaf page.
You can obtain Double Magic early by picking up the deku leaf with storage, reloading the area, and picking up the deku leaf again.
It is possible to enter the waterfall where Makar is located before the Wind Temple by using only a jumpslash. This is particularly useful if you want to grab forest water first, as you can then follow the water down and use a late jumpslash to bypass the waterfall. Though it is possible to also jump from the platform near the post and get behind the waterfall with a jumpslash from there, it would be faster to simply grapple the post from bellow.
From the lowest Forest Haven loading zone, use the grappling hook to swing. Press A to let go when you are at your highest point and leaf to the chest. A few leaf pumps are necessary.
See the Forbidden Woods page.
After getting storage using Beedle Ship near the Forest Haven entrance (where the mailbox here), you can use a specific setup in order to charge up a superswim (using camera lock) and get an air refill from Beedle Ship. This is useful in order to swim all the way from Forest Haven to Outset without any pitstop for exemple.