I think i caught the whole plot of this manga.
First the white-haired girl didn't kill the MC's father. Instead, she made it look like it so the MC's can continue his father's work (chasing the real culprit) through the motivation of killing her. Because there's a catchphrase that seems to be a major spoiler of this manga. The father saying about "believing in others" and the first case of this manga about a husband who is suspected of having an affair but in reality he is not having an affair, even though the evidence collected by the MC seems to support that the husband was having an affair. So I see a glimpse of the author's hints about how this manga is going ended. The moral story is "do not immediately believe to what it seems to be and trust the others". The white haired girl was lying to MC and the lie will revealed at some point of the story and *bam* they cooperate and catch the real culprit OR the MC, by his foolishness, killed her and knows the real meaning of her lie soon after that. Too bad, I'm afraid it would be like that because the MC turns into some edgy-shit protagonist and controlled by anger and cannot see the meaning of the woman's actions. The white haired girl could not possibly kill the MC's father because she respected him so much and she is very cares about MC and maybe she loves him too.
Sorry for my bad grammars. And remember, this is just some my speculation. Don't know if this might came true!! Feel free to roast me if it didn't turn like this haha.