Staying some time in the Schengen area after your current visa expires is possible as the time under the work-holiday visa does not count toward the 90-day limit, see Does tourist visa (90 days) apply after a working holiday visa ends in Schengen countries?
Applying for a work-holiday visa from another country should also be no problem, at least as far as Schengen regulations are concerned. The basic principle is that long-stay visas are still national matters, not subject to the Schengen-wide restrictions on short stay (“Schengen visa” is a bit of a misnomer in this case, there are major differences between Schengen uniform short-stay visas and national long-term visas from Schengen countries).
If you are eligible under local law, a Schengen member state can certainly grant you a national visa starting immediately after another long-term national visa (from the same country or from another Schengen country). One thing you need to be mindful of is that it's sometimes impossible to apply for a long-stay visa from within the country (even if you would otherwise qualify for the visa) but some countries do allow it.
Importantly, none of this is true for short-stay visas (Schengen uniform/type C visas). In that case, you shouldn't have overlapping visas and even in the unlikely event you would manage to get several visas, the 90-day rule applies to you as a person. Getting another visa or using another passport does not entitle you to another 90 days in the Schengen area.