I have never heard of an oil removing hyperpigmentation. I use an AHA peel for that.
I use jojoba oil on my cuticles. It is very similar to your natural skin oils. If you don't get a lot of acne from your natural skin oils, try this. It can be very helpful for those with dry skin, but not oily skin.
Grapeseed oil is good for cleansing.
I've used argan oil, and it made my oily skin more oily but feed dry and tight at the same time. It works better for my hair, not much else.
Honestly i really like coconut oil as a moisturizer. I have oily skin, and a bit of it goes a long way. I use it on my very dry areas, and it helps my skin heal without looking like an oil slick.