Upvote:5
Timatic, the database used by airline staff states the following:
The following are exempt from holding a visa:
Passengers with a double or multiple entry "C" visa, valid for all Schengen Member States . The visa must be valid for the period of intended stay. They are visa exempt for a maximum stay of 90 days.
So, as you have the Schengen Visa and is is valid for a full month this is more than enough to cover your entire trip, you don't need anything else.