The big Island has a very large population. There has to be meetup groups dedicated to things that might interest you such as hiking, spearfishing, etc. Spearboard.com is a site I use in california, they have a handful of Hawaii fisherman on there. I'm sure there are equivalent forums for fishing or kayaking if you're not into spearfishing.
You're probably near the lava tubes. Not a great way to meet people probably, but worth checking out while you're there. There are other cool things worth seeing, but I would say that is the most unique sight in hawaii. I'm also into coffee so I checked out a few coffee plantations.
Also, just invite a member of your unit to go do something and split the cost of renting a car. Worst case they say no, but somebody is bound to be interested at some point. Unless the issue is that you're an officer and they're all enlisted, so you're not supposed to mingle. I knew a guy in the virgin islands that had that issue, I felt bad for him, he had no peers at his base to hang out with.
Hitchhiking is a reasonable option in hawaii, works much better than in the continental states.
It sound like so far you haven't made any attempt. If you put yourself out there on something like meetup, you might end up in an occasional awkward situation hanging out with a stranger you don't have much in common with, but that's better than sitting in your barracks doing nothing, and you'll end up with some fun times sprinkled in there with people you do get along with.
We live in a weird age where people tend to be very insular, and yet if you're not interested in that, these tools like meetup really do work to meet some compatible people and make some new friends.
You also live near some hostels, spend a night or two in one and you'll probably meet some other travelers who are looking for people to hang out with as well. I met people at a hostel in Hilo and Kona that way, by just saying hi to people. Some of the hostels even organize groups to go out on buses to various locations.
Seriously, there is no excuse for not finding something fun with somebody you like. Maybe you won't make a lot of really significant friendships in the next four months, but you're bound to at least make some casual friends in that amount of time.
Make a specific plan one night a week to look at nearby meetup events and pick two to make it to each weekend. Pretty soon you'll wonder why you thought it seemed so hard.