Can I apply for a longer Schengen visa to overwrite my shorter visa?

I have recently been approved for a Schengen visa to France for 15 days. I found the means and the desire to stay in France for about 30 days, but the visa only allows 15 days. The difficulty is that the period of validity is 30 days, but the duration of the stay is only 15 days. This makes it impossible to apply for a Schengen visa from the second country for the duration of my stay there, after the 15 days of the first one have elapsed.
My question: I have heard that it is possible to apply for another Schengen visa and have the first one canceled. Can I request one that overlaps the previous one and have a reasonable expectation that it will be met?