A student visa for more than 3 months (which I assume you will need to attend a full term) is not a Schengen short-stay visa but a national long-stay visa. Those are sometimes called “Schengen type D visas” but that's a bit misleading because they have not been unified and they are not issued based on the Schengen regulations. There are actually hundreds of different type D visas, all with slightly different rules.
I don't know the rules in the Czech Republic but you are unlikely to be able to extend a student visa if you do not intend to continue your studies in the Czech Republic. Incidentally, in other countries, student visas typically expire in September, not in June, to give students an opportunity to stay in the country until the beginning of the next term but, again, I don't know whether that's the case in the Czech Republic, especially for a relatively short visa.
The thing is that extending a visa only makes sense if the premise for your stay hasn't changed, if you want to do something else, you need a new visa. Because you don't have a Schengen visa, what you will need to do if you want to visit the Schengen area after the expiration of your student visa is therefore a fresh visa application. You need to apply in the country where you presently reside (and I think you should be able to apply in the Czech Republic during your time there) but to the consulate of the country that will be your main destination.
If you plan to work as a volunteer somewhere then definitely apply to that country and no other. But don't go there to apply, apply from where you are, at a consulate, before entering the country. What you will get if successful is simply a new visa (of a different kind), not an extension.
Finally, a Schengen visa application requires different documents, do inquire about that now to be able to take what you need with you to the Czech Republic. You can then lodge your application from the Czech Republic, approximately three months before the end of your Czech visa/the beginning of your stay in the other country.