I don't think a light oil would be much of a problem. I mean you get oils on your scalp when you apply them to your hair a bit anyway. I'm not sure I see why you would do that unless you had dry skin on your scalp.
It probably doesn't have to be a gel, but yeah generally speaking a product with hold will help with second day hair. I know mahoganycurls on YT uses a harder hold gel than she would like, Ecostyler, to get hair that lasts all week. I had to find a balance between hold and softness. I don't like crunchy hair and crunchy hair actually looks bad on me the next day. I think it's because my hair is shorter, so it mats up a bit and I need that flexibility. If I could pineapple it might be different.