Even when it comes to applying for a Schengen visa, we would advice you to apply through the embassies of the bigger countries: namely, France, Switzerland or Germany. There processing time is less, they have embassies in most metropolitan cities, and they are generally more professional. Unfortunately, that will mean that at least one of these countries should be a part of your itinerary. Its a bitch, but then again, they are great countries with so much to offer. Each time to you visit them, you will find something new to discover!
A large database called theis shared between all Schengen countries for purpose of background checks of the visa applicants. If you re-apply for visa through another country within short duration of visa rejection, it can raise red flags against you!