How do I open the chest on the southeast corner of Tikarel Island?

Go just on the cliff above, there is a crate (the same crate you pushed next to the chest I guess). Push the crate and make it fall, then jump on top of it. No need to touch the crate more, just leave it. A mere last jump will do the trick.

You don't need the boots to get this. From the top cliff, push the box back toward the well, then off the cliff. It's hard to describe, so below is a diagram of the smaller cliff below. The O's are the bits of land on the smaller cliff, and the X is your target.

_ __ O O

O X O O

You want to put the box in the space where you can push it from the left over to the bigger open spot. If you push the box down and then to the far right (so it lights up with the chest), you can actually walk through the two rocks that look like they block you. The chest gets you a chunk of XP.