I already have a Schengen Visa and will return home in March.
Can I apply for a Schengen visa again in April and come back in April for another 3 months? Or is there a way to extend Schengen Visa for another 3 months?
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Visa and length of stay are two separate (albeit obviously related) issues. Short stays in the Schengen area are limited to 90 days in any 180-day period (see the relevant discussion) so if you stay there for three months, you need to be somewhere else for three more months before you can come back on a short-stay visa.
Depending on your exact situation, you might or might not be able to get a new Schengen visa but it will not open any right to stay longer. Some people can visit the Schengen area without visa, others have short-stay visas valid for several years but they all have to abide by this rule. It also doesn't matter which country issued the visa or if you get a new short-stay visa from a different country than the one that issued your first visa.
The only way to stay longer legally is to obtain a national long-stay (type D) visa or, in exceptional cases, a limited territorial validity visa (and exceptional means exceptional, it's intended for diplomats participating in talks at the UN and these sorts of things).
If you do want to apply for a new visa, two things are worth noting: