Multiple Entry Visa Cancellation and Applying New Visa [duplicate]

You do not need to cancel your visa.

The Schengen Visa Code, Article 9(1), explicitly contemplates applications for multiple-entry visas during the validity of another multiple-entry visa, as long as the validity of the first visa is at least six months:

CHAPTER II

Application

Article 9

Practical modalities for lodging an application

1. Applications shall be lodged no more than three months before the start of the intended visit. Holders of a multiple-entry visa may lodge the application before the expiry of the visa valid for a period of at least six months.

This is also reflected in the Handbook for the processing of visa applications and the modification of issued visas, in Part II ("Operational Instructions on the Processing of Visa Applications"), section 3.1 (the last example and the preceding paragraph are relevant to your case):

3. LODGING OF AN APPLICATION

   3.1. When can an application be lodged?
   Legal basis: Visa Code, Article 9(1)

   A visa application should in principle be lodged at least 15 calendar days before the intended visit (as this is the normal processing time) and cannot be lodged earlier than three months before the start of the intended visit. It is the applicant's responsibility to take the necessary precautions to respect the deadlines where an appointment system is in place. However, applicants should be informed of the various deadlines, see the Handbook for the organisation of visa sections and local Schengen cooperation, Part I, point 4.

   An application lodged less than 15 calendar days before the intended departure may be accepted, but the applicant should be informed that the processing time may be of up to 15 calendar days. If, nevertheless the applicant insists on lodging the application he should be informed that the final decision might be taken after the intended date of departure

   Example: A Turkish national decides to book a special last minute offer for a skiing vacation in Austria with a departure within two days and only the day before departure he realises that he needs a visa to enter Austria.

   In this case the Austrian consulate could refuse to deal with the application.

   A holder of a multiple-entry visa may apply for a new visa before the expiry of the validity of the visa currently held. However, the validity of the new visa must complement the current visa, i.e. a person cannot hold two uniform visas valid for the same period in time.

   Example: A Moroccan lawyer representing a gender equality NGO who frequently participates in meetings in various Member States holds a multiple-entry-visa which expires on 31.5. She applies for a new visa on 15.4.

   If a new visa is issued, it should be valid from 1.6. and in such a case the visa holder would be entitled to enter the territory of the Member States on the basis of the first visa that will expire during the stay and leave on the basis of the new visa.

The last example clearly shows that you should be able to submit an application for a multiple-entry visa during the validity period of another multiple-entry visa. If the second visa is issued, its validity must begin no earlier than the day after the expiration of the first visa. So the Italian consulate (and its third-party visa service provider) should indeed process your application for a visa beginning on June 9th.