Hop on a Ferry to Molokai for Deer, Pig, or Turkey... No season and no bag limit on the first two. Axis dear where first introduced to Molokai as a gift from the British to King Kamehameha V for subsistence. There are like 10,000 deer on that island and only 14,000 people (heard that once). But it's true they are everywhere.... I have a few contacts that can get you a rifle and a couple more that would let you hunt private for a small trespass fee. If (when) I go back to hunt Hawaii, it will be on Molokai every time.
Also ridiculously good bone fishing and I know the best spear fisherman on Molokai.
Note: "this is the least friendly island to Haoles" but I didn't experience that when ever I went there.
Note II: Hawaii has some strict gun laws, do some research before you go. Less troublesome as a tourist but they have them.