In North Carolina, the governor vetoed efforts by lawmakers to reopen skating rinks, bowling alleys and amusement parks. In Alaska, new workplace clusters are emerging, social distancing is on the decline and contact tracers are overwhelmed. And in Kansas, state and local leaders are squabbling over whether masks are required.

“Early on, people who tested positive usually had a short list of close contacts,” Dr. Joe McLaughlin, Alaska’s state epidemiologist, said in a statement. “Now, as people are mixing more with others, it’s not uncommon for someone who tests positive to have had dozens of close contacts, sometimes too many to name and call.”

The struggles in those three states, all of which set single-day case records on Friday, exemplify the challenges officials across the country face as cases surge. Unlike the first spike in March and April, when most places were on lockdown, case numbers are now exploding after many Americans have returned to their routines and grown frustrated with restrictions.

In Kansas, where more than 780 cases were announced on Friday, residents have heard mixed messages from their leaders. This week, Gov. Laura Kelly, a Democrat, ordered residents to wear masks in public. Commissioners in Sedgwick County, which includes Wichita, then voted to make Ms. Kelly’s mask mandate a recommendation, not a requirement. But on Friday, the Wichita City Council convened in a special meeting and approved a mask mandate, effective immediately, with the possibility of fines for those who refuse.

“We have a shot of avoiding another shut down, of ensuring our kids have school & protecting folks,” Mayor Brandon Whipple of Wichita said on Twitter after the city’s mask rule was approved.

Similar whiplash was seen in North Carolina, where the Republican-controlled Legislature passed bills that would have curtailed business restrictions enacted by Gov. Roy Cooper, a Democrat. But Mr. Cooper stepped in and vetoed the measures, meaning roller skating rinks and bowling alleys, along with some other businesses, must remain closed.

“Opening these higher-risk facilities would spread Covid-19 and endanger the state’s flexibility to open the public schools,” Mr. Cooper said in a veto statement.