Telling someone who has oily skin they need to moisturize may sound silly but thats exactly it. Even the oiliest skin needs moisture thus, skipping oil can lead to skin that is chronically dry which is why skin starts to produce more oil.
Many with acne avoid oils, but there are some oils like argan and coconut oil that are compatible with the skin, absorbs well, and even work [to prevent] acne. These oils have antibacterial properties that help balance the skin. It may seem strange to slather on some coconut or argan oil over your broken-out skin, but it may just do the trick.