In the 12 years that Michael R. Bloomberg spent as mayor of New York, he became known for pushing far-reaching health initiatives to curb smoking and change eating habits — challenging Big Tobacco, big sodas and, at times, the will of New Yorkers.

The first three years of Bill de Blasio’s tenure as mayor have brought a different approach.

Mr. de Blasio has championed other types of public health initiatives — most notably to promote mental health — but for the most part, he has not advanced Mr. Bloomberg’s legacy.

Nowhere is that more apparent than in the city’s approach to curbing tobacco use.

“Clearly the evidence is still there that tobacco is a big problem in New York City, and it’s also clear there are more things we could be doing,” said Kevin O’Flaherty, the director of advocacy for the Northeast region for the Campaign for Tobacco-Free Kids, “so I’m at a loss to give a logical reason for why these things aren’t being addressed as aggressively perhaps as they were in the past.”

Anti-tobacco advocates point to a series of potential initiatives, some of them already taken in other cities, that could extend New York’s fight against tobacco use. They include measures to limit the number of stores where cigarettes are sold, increase taxes on small cigars and other cheaper types of tobacco, ban tobacco sales in pharmacies and put a brake on the proliferation of hookah bars.