I have seen issues like this however it was cooking oils that spilled on the phone, but same concept. If i were you i would take the entire phone apart and clean the mother board off, taking any EMI shields off that you can manually, and look to see if any chips have oil on them, if so wipe them off, and i would also take the screen off and wipe down the back of the backlight, the oil will move around and can get into your backlight and then you will have light and dark spots on your screen. I would also check to see what the hair oil is made of, if it have any water in it i would also recommend buying at least 91% isopropyl alcohol which you can get at rite aid, pour a little bit of alcohol out onto a table or in a little up and use a tooth brush to dab it into the alcohol and then gently scrub the board. This will cause any water to evaporate. Also if you think it is necessary wipe down the inside of the phone with a paper towel that has some alcohol on it. Once dry put it back together and see if it works. If it doesn't work you may need to replace the audio flex (which has the vol. buttons, silent switch, noise canceling mic and headphone jack on it) and the ear speaker. (If you do happen to buy these parts, for the ear speaker just replace it with the new one, and for the audio flex just plug it into the board and test to see if it works (test all functionality including phone calls, if it works then you know you have a working phone and can proceed to install the new flex, if not then you know that you did not just spend all that time installing it for nothing). If it still does not work then i am not sure if it is worth sending out to get fixed depending on when your contract is up and you can get a new phone, sometimes you can find shops that are good with fixing water damage issues when they have trained techs and/or that are actual engineers, but it may be a bit pricey.
I wish you luck in getting your phone fully functional again!