Your delayed kill means that Lookouts will have a harder time finding you. You can use this to "prove" your innocence in the latter part of the game. However, smarter Lookouts will notice you visiting your target, so when they die a day later, they will be suspicious of you or even call you out if their target claims they were poisoned.
Doctors are a high-priority target; with them in play, you are virtually useless until you have the Necronomicon, reason being that anyone poisoned can simply claim so and the Doctor will heal them. Because of this, try to poison Jesters on night one, whom Doctors won't save.
Claiming to be poisoned has its benefits - it means the Doctor will be healing you that night, protecting you from harm. Even if the Town notices you haven't died, you can simply claim to have been healed. Of course, this will not work once the Coven gets the Necronomicon, as healing no longer prevents death by poison.
Coordinate with your fellow Coven members on who you should poison. Before night 3, you might be the only Coven member who can actually kill, with the other early game killer of the Coven being the Potion Master.
If you play with a Medusa, you can ask her to claim poisoned while you don't poison anyone and ask her to stone visitors the following night. This might allow you to kill the Doctor and make your poison incurable.
As a Lookout, when someone dies to poison, if you were watching them last night, anyone who visited them could be the Poisoner.
As a Doctor, make sure to heal anyone claiming to be poisoned. Not only will it purge them, it will protect them from any killers. On the other hand, if you believe the Poisoner has the Necronomicon then healing poisoned people is a waste of a night.
As a killing role, targeting people who are poisoned is a bad idea. Not only will they likely be protected by Doctors, even if they are defenseless, they will die from poison anyway, wasting a kill.
As a Tracker, you can call out a Poisoner if you see them visiting someone who claims to be poisoned the next day.