A common misconception. Below quotes are from https://www.bls.gov/cps/cps_htgm.htm, bolding mine. Also, the BLS report has six different measures of unemployment counting various different groups. The "headline number" of U3 counts only people actively looking for work, but other measures count people who want jobs but aren't looking and people who work part time but want to be full time. Labor force participation is also in there, counting people even if they don't want to find a job. It's all reported!
Some people think that to get these figures on unemployment, the government uses the number of people collecting unemployment insurance (UI) benefits under state or federal government programs. But some people are still jobless when their benefits run out, and many more are not eligible at all or delay or never apply for benefits. So, quite clearly, UI information cannot be used as a source for complete information on the number of unemployed.
Because unemployment insurance records relate only to people who have applied for such benefits, and since it is impractical to count every unemployed person each month, the government conducts a monthly survey called the Current Population Survey (CPS) to measure the extent of unemployment in the country. The CPS has been conducted in the United States every month since 1940, when it began as a Work Projects Administration program. In 1942, the U.S. Census Bureau took over responsibility for the CPS. The survey has been expanded and modified several times since then. In 1994, for instance, the CPS underwent a major redesign in order to computerize the interview process as well as to obtain more comprehensive and relevant information.
There are about 60,000 eligible households in the sample for this survey. This translates into approximately 110,000 individuals each month, a large sample compared to public opinion surveys, which usually cover fewer than 2,000 people. The CPS sample is selected so as to be representative of the entire population of the United States