Separate Visa from embassy after Schengen visit [closed]

US citizens cannot get a Schengen visa in any event and nobody can get a (regular type C) Schengen visa if they used up their 90 days of stay in the area. There is no legal way to get a visa that would allow you to stay anywhere in the Schengen area for more than 90 days but there are several ways to get the right to stay in a particular Schengen country.

The relevant regulation does provide for something called a “limited territorial validity” visa (LTV) that allows the holder to stay in a Schengen member state for more than 90 days. LTV visas are restricted to that specific member state (or, possibly, several member states) but they can allow the holder to stay more than 90 days in the area. They are intended for humanitarian or other limited purposes and most likely won't be available to you.

More realistically, you could perhaps get a national long-stay visa (in Schengen parlance a “type D” visa), which some – but not all – countries do issue for visits and tourism. That visa will only be valid for one country but it won't fall under the 90-days-in-any-180-days rule. You should therefore try to see if something like that is available for your destination of choice. In some countries, a student visa might also be a realistic option (applying for one without actually intending to study would obviously be a form a fraud, even though it's often possible to get away with it, at least once).

Finally, in limited cases, preexisting bilateral agreements allow the nationals of specific countries to stay in some member states even if they already used up their 90-day maximum stay in other countries of the Schengen area. For example, New Zealand has bilateral agreements with a dozen countries including Germany, France, Italy and Spain…

Unfortunately for you, it does not seem to be possible in many places for US citizens but it is at least possible in Denmark. To benefit from this and other similar rules, your previous stay must have been in another country than Denmark (i.e. you can stay 90 days elsewhere in the Schengen area + 90 days in Denmark but not 180 days in Denmark).