That only makes sense in places where the taxes go to healthcare organizations. In the American system, that doesn't happen - there's no way that a city will hand the proceeds of its tobacco taxes to insurance companies.
Tobacco companies already pay states a bunch of money from their 1998 settlement.
https://en.wikipedia.org/wiki/Tobacco_Master_Settlement_Agre...
It's over $200 billion and growing. They're more than compensated.
Cigarettes and tobacco products are already heavily taxed for the user. Both the federal government and the state government makes a lot of money off people's addiction, and it's an addiction they can't break (obviously).
https://igentax.com/cigarette-tax-state/
Time to leave smokers alone. Many if not most of them got hooked as kids and as a society we just keep punishing them for their "moral failure."
Increasing the price isn't necessarily about making it inaccessible to people so they stop smoking (though it is a minor part of it that is a nice side-effect), it is to make sure that the additional burden placed on the healthcare system by those people is balanced out by taxes paid in form of increased prices on tobacco products.
If people don't stop smoking, then might as well make sure they cover the extra cost they add to the healthcare system. Seems pretty fair.
To be clear, this statement I made has nothing to with my views on the specific menthol ban in question, as I am still not sure how I feel about it. I was just discussing the hypothetical non-existent strategy of just more taxing on tobacco products instead of outright banning them.