Getting a vampire in the first couple of moves always feels a bit unfair, especially when it happens a few times in a row. Unless I'm misunderstanding the gameplay, there's nothing deductive you can do until you've filled out a column/row to get a hint.
I wonder if there's anything that could be done to give the player the occasional small edge in that scenario. For instance, perhaps a random unused hint could carry over into the next round? Or perhaps a parallel progression system/shop could let you trade some blood to tilt the odds gently towards you? (Roulette Knight does this pretty well)