Any small molecule fatty acid can be comedogenic, otherwise if you are using virgin coconut oil it might be 'contaminating' proteins that are causing the issue so a refined oil might be better. Otherwise you could try a different source of lauric acid - babassu oil, palm kernel oil, tucuma butter.
Try cutting sulphate surfactants out of your skincare routine because these are proven to thin and dehydrate the skin at concentrations as low as 1%. Instead use a shower gel that respects the skin's protective acid mantle/ beneficial skin flora (pH 4.5 to 5.5) and perhaps moisturise with a product containing medical grade lanolin.
Also review your diet so that it is as anti inflammatory and nutrient dense as possible (esp. oily fish). You'd be surprised how much you can strengthen your skin's barrier function with lifestyle modification. You need to be patient with this, it may well take a few weeks for product changes and nutrition combined to make a difference.
If the coconut oil doesn't break out your scalp and hands just keep your hair away from your skin. Do EVERYTHING upside down, cover your neck with saran wrap and shoulders with a hairstyling cape, apply the oil to the ends of the hair only and secure, pop on a shower cap, wash out the excess upside down.