If its a tiny island I just sail around it. A large Island I dock on the coast, build a workbench and repair the boat then build a portal and head back to base. I get my rested bonus then run along the coast in one direction until half my rested bonus is gone then turn and go interior parallel to the coast until I get back to the boat and portal.

Then I rest again and do the same thing going along the coast in the opposite direction. If that hasn't cleared the island I pick up the portal and the work bench and sail along the coast that I had explored until I get to unexplored area. Then I dock, build, and repeat until I have the coast and most of the exterior explored. Then I fill in any remaining interior.

After that I exploit any resources that I need that are found there, put the ore(s) in the ship and portal every thing else back. When the ship is full I sail it back to my main base, unload it, then destroy the boat to recover the materials, and portal back to the island that I just finished. Then I rebuild the boat. take the portal down, then the work bench, and sail to the next island.