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or if she can just stay in Italy (she is way below 90/180 days).
A new multiple-entry visa [MEV] should always start directly after the previous one, if issued before the first one expires.
The sample given in Visa Code Handbook covers your friends situation exactly.
She can enter with the first and leave with the second as long as the 90 days is not exceeded.
Visa Code Handbook, Chapter 8.3 (page 77 of 2020 pdf)
Holders of a valid MEV may be issued a new multiple-entry visa with a long validity provided that the validity of the new visa starts at the expiry of the previous one and the existing valid visa should not be revoked. Responsibility for complying with the 90/180-day rule lies with the visa holder.
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.