1. Travel with Both Documents: When travelling with a Schengen visa that has your previous name, it's advisable to carry your marriage certificate (or other legal documentation of the name change) along with your passport. This can help avoid confusion or delays at border control, but to be honest, it is too risky. The regulations covered this case are pretty misty and no guaranty you won't face serious difficulties on the border.
2. Notify Issuing Authority: While it's not always necessary, some individuals prefer to inform the embassy or consulate that issued their Schengen visa about the name change. You can contact the issuing authority for guidance on whether they recommend any specific steps in this situation.
3. Consider Applying for a New Visa: If your current visa is close to expiration or if you anticipate needing a new visa for other reasons (such as a more extended stay or a different purpose of travel), you may choose to apply for a new Schengen visa with your updated surname. In this case, you would cancel the existing visa and follow the standard visa application process, providing all required documents, including your updated passport and marriage certificate.