Сложный визовый вопрос! Поездка по двум стыковым визам и окончание первой визы в поездке

Да, это возможно.

Пруфстатья:

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.

Источник: http://ec.europa.eu/home-affairs/policies/borders/docs/c_2010_1620_en.pdf 

Это не просто пдфка, это офицальный документ: COMMISSION DECISION 

of 19.3.2010  establishing the Handbook for the processing of visa applications and the modification 

of issued visas  (Only the Bulgarian, Czech, Dutch, Estonian, Finnish, French, German, Greek, 

Hungarian, Italian, Latvian, Lithuanian, Maltese, Polish, Portuguese, Romanian, 

Slovak, Slovenian, Spanish and Swedish texts are authentic)