Staying 90 days after your current visa expired seems complicated but applying for another work-holiday visa should be no problem, at least as far as Schengen regulations are concerned. The basic principle is that long-term visas are still national matters, not subject to the Schengen-wide restrictions on short stay (“Schengen visa” is a bit of a misnomer, 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.
In principle, I think that you can also go to another Schengen country (for up to 90 days in total in any 180-day period and obviously without working there) under your current long-stay visa. So the best thing to do is to apply for a new visa before the end of your current visa instead of counting on a visa-free short-stay to bridge the gap. You should probably do that in advance to leave some time for your application to be processed and plan your next move if your next visa is refused.
Importantly, none of this is true for short-stay visas (Schengen type C visas). In that case, you shouldn't have more than one visa (formally, if you need to apply for a Schengen visa and you already have a valid one, you should first have the country that delivered that first visa invalidate it before applying for the next one) and even in the unlikely event you would manage to get several visas, the rules apply to a person.