Upvote:1
Getting a new visa that starts right after the previous one expired is perfectly possible. There is nothing in the regulation preventing it and it's actually among the βbest practicesβ recommended by the EU Commission for multiple-entry visas. It's also fine to enter the Schengen area on one visa and leave on another one, as long as there is no gap between them, see Traveling on back-to-back Schengen visas
My readings of the rules is that yours is not a situation in which they would typically extend a visa but I am not completely sure about that. A consulate can also of course cancel a visa it issued and replace it with another one but, unless it's to correct a mistake they have made, they will probably require that you submit a new application and pay the fee so they might just as well issue a separate visa.
Beyond that, whether you will actually get a long multiple-entry visa might depend on many details (your profile, past travels to the Schengen area, location, consulate/destination, the documentation you submit/need to travel).
Also, two notes about the procedure: