Upvote:5
No you do not need another visa. Schengen visa gives you access to all Schengen area so you are free to move within Schengen area as long as it is valid and you follow the 90/180 rule regardless of the issuance state and the entry state. In other words think of Schengen area as a big country and think of member states as cities of that big country, you are granted access to that big country so visiting cities within that big country is granted by default.