This diagram in my textbook illustrates two solutions to a negative externality of consumption - here, the consumption of cigarettes. It states that MSC is shifted upwards by the specific tax. Since the deadweight/welfare loss is defined as the area demarcated by MSC, MSB and the quantity demanded, this would seem to imply that the deadweight loss is not reduced at all (as per the diagram). This seems rather strange.
Here's an alternate diagram I think makes more sense.
Since an increase in taxes will effect a change in the behaviour of individuals, I would argue that it is the MPC (and not MSC) curve that is affected by taxes (hand-drawn Diagram 1). Diagram 2 shows the same picture, but using an ad valorem tax. This will cause a reduction in the deadweight loss. One could argue that a tax will also cause an increase in MSC (effects on employment of tobacco farmers etc.), but, by my reasoning, for the policy to be effective ΔMPC>ΔMPC (otherwise the welfare loss will not be reduced). See Diagrams 3 and (w/ ad valorem) 4 for this scenario.
Does this make sense?