This is what I think an increase in "involuntary" unemployment looks like:

The vertical axis represents the quality of a job, in the eyes of someone searching (or simply waiting) for a job. "Quality" will presumably include wage, but many other characteristics as well.

The horizontal axis represents the expected duration of searching (or waiting) to get a job of that quality.

The blue curve represents a trade-off. The better the job you want to get, the longer you expect to search (or wait) before you get it. (But the trade-off probably asymptotes to some maximum, or maybe even declines if you wait too long and your skill human capital depreciates.) You pick the tangency point where your indifference curve kisses the trade-off. (Rather, you aim to pick that point, because you don't know what the trade-off will really be, and you might get lucky or unlucky.)

Now suppose the economy enters a recession. The trade-off worsens to something like the red curve. You choose a new point. You will probably choose a point as drawn, expecting to search or wait longer and get a worse quality job.

We could argue about whether the unemployment in the first equilibrium is "voluntary" or "involuntary", but there doesn't seem to be much point. We could even argue about whether the increased duration of unemployment when we move from the first to the second equilibrium is "voluntary" or "involuntary", but there doesn't seem to be much point. In both cases, a person with different preferences would have chosen differently, so you could say it's "voluntary". But the worsening of the trade-off was definitely involuntary. The unemployed person has no choice about that worsening trade-off; only whether to respond to it by accepting a worse job, or being unemployed for longer, or a bit of both.

And whether that worsening trade-off was the result of an efficient response of the economy to some real shock, or the result of an inefficient response to some daft monetary policy shock, is a quite separate question.

Update: And how that trade-off facing an individual worker would shift if all workers changed their preferences at the same time is also a quite separate question. It might shift in or it might shift out. Beware the fallacy of composition. That's why God invented macroeconomists.

In very simple supply and demand models of the labour market, the trade-off looks very different. All jobs are the same with one wage. The trade-off is L-shaped (if you twist the L clockwise through 90 degrees). If the wage is at the market-clearing level, the trade-off is vertical at 0, then goes horizontal at W. If the wage is above the market-clearing level, the L shifts right, so you have to wait for U periods, then get a job at W. If you would refuse a job at W, your unemployment is voluntary; if you would accept a job at W, your unemployment is involuntary.

But the labour market is more like the marriage market. Heterogeneity of jobs and workers matters. If people didn't care who they married, we could divide people into voluntarily single and involuntarily single. But they do care, so we can't. But we can talk about worsening trade-offs.

Just adding my twopenceworth to the debate between Roger Farmer and David Andolfatto (and Roger again). Three-way Western wars.

Update: we could draw a similar trade off for employers with vacancies, with quality of the worker on the vertical axis. But in a recession the trade-off for the employer improves. It's easier to find a better worker quicker for lower wages. Because recessions are a monetary phenomenon, where those who want to sell illiquid goods and buy money find it harder to do so, and those who want to buy illiquid goods and sell money find it easier to do so.