For people with attitudinal conflict, direct persuasion may not necessarily be effective. As Maliszewski (2011) highlights, people respond to conflict between their internal and external attitudes by exaggerating the latter to overcompensate. As a result, despite agreeing with anti-smoking messaging internally, smokers would be driven to ignore it and light a cigarette faster. Persuasion would have to be directed at identifying the internal attitude and exposing it to the person to succeed. One possible approach that may be effective is to not start with the main point of contention immediately, as there is a considerable chance that the person will immediately adopt a defensive posture based on their external attitude. Instead, beginning with matters that relate to the internal attitude and gradually transitioning toward the main topic may be a more effective approach, though it has to be done very carefully.
Another potential way of persuading people with attitudinal conflict would be to distract them with another matter during the discussion. As Maliszewski (2011) notes, the splitting of cognitive resources makes people more likely to apply their intrinsic attitudes over their explicit ones. With that said, once they have some time to reflect on the arguments, they can apply their external attitude again and reject the provided points. Still, this approach provides an opportunity to deliver the point to the person and potentially reinforce it later on. It may be considered somewhat unethical to attempt to persuade a person while they are distracted by something else. However, as long as they can consider the arguments in a more relaxed setting later and the attempts at persuasion are not overly aggressive, the method should not be problematic as a means of delivering the initial message.
Maliszewski, N. (2011). How do people resolve conflict between implicit and explicit attitudes? Polish Psychological Bulletin, 42(1), 36-45.