The measures are available, and they're usually in the vicinity of twice the regularly-reported (U-3) rate.
Several of them do measure underemployment.
The government and media largely rely on U-3 (IIRC) when reporting unemployment. The upside is consistency - they all use the same number. The downside is this figure may not be fully representative of the economy.
I would argue that U-5 or U-6 better represent the state of the economy (as it relates to employment), as both included some set of under-employed and recently-given-up would-be workers.
But, we'd have to make a concerted effort to cut over to those numbers to avoid confusion.
Examples:
"How bad is our current recession compared to the great depression for the workforce?"
"How might raising the top marginal income tax bracket to 70% affect the labor market?"
"How does relaxing strict immigration policies contribute to job loss in a nation (if at all)?"
But, given an increasing income disparity between the top ~5% and bottom half, and a continuing sense of "being left behind" by many workers in labor-intensive industries, it seems doing a better job of describing those disenchanted workers (whether they be part-time at AutoZone, or recently given up looking, etc) would be beneficial to all.
For one of those disenchanted employees, seeing "Unemployment is at all-time lows! The DOW is at a record high!" on the evening news (while they're working 3 part-time jobs for $9/hour and zero benefits) isn't just unhelpful, it's insulting.
A lot of comments here are heavily biased towards accurately accounting for the retirement end of life, without considering the other end - people who are intentionally not working or are under-employed for some other reason. The one I had mainly in mind was highschool and college-aged students who are working part-time: they'd get included in under-employment, but aren't necessarily looking for a living wage yet because they have other priorities, so they don't get included in unemployment counts.
Also,
> I'd like to see unemployment measured more like total population minus those employed.
This measure would include children who can't even work part-time.