I have business Schengen visa issued by Italian embassy from 15th July to 22 Oct 2019. I am going to Italy for office work and coming back on 10th Aug to India. Then on 10th Sept I am going to Europe on vacation with my family. I have applied for tourist visa with French embassy for my family members since longest stay will be in Paris. We are planning to visit Paris, Zurich, Venice, Salzburg & Munich. We will come back on 24 Sept to India.
Do I need to apply for another schengen visa or is my multiple entry Schengen visa enough?