LINCOLN, Neb.—Full employment has arrived in this Midwestern capital, offering a taste of the higher pay and new perks more companies are using to lure workers in a tight labor market.

At 2.3% in the most recent reading, Lincoln had one of the lowest jobless rates in the country. It’s one of 31 U.S. metropolitan areas—from Portsmouth, N.H., to Provo, Utah—with unemployment rates at or below 3%, far better than the 5% average nationwide. A year earlier there were only 18 such spots, highlighting steady improvement in the labor...