The requirement will remain in place for congregate care facilities.
The state's positivity rate, which measures the proportion of tests returning positive, dropped to 3.07% on Saturday, a decrease from 3.40%.
Gov. Andy Beshear said it's "too early" to name the facility, and health officials have not yet determined the scope of the variant.
The judge issued orders blocking the three bills pending further proceedings in the high-stakes case.
State health officials on Saturday reported 46 more deaths in which the coronavirus was a contributing factor, alongside 2,649 new cases of the respiratory virus.
More than 3,300 Kentuckians have died from the virus since it was first reported in the state.
As of Sunday, 1,713 Kentuckians were hospitalized with COVID-19, Gov. Andy Beshear said.
As of Wednesday, 1,673 Kentuckians were hospitalized with COVID-19, according to a news release from the governor.
Of the new cases reported Wednesday, 415 were confirmed in Jefferson County, Kentucky Public Health said in its daily report.
As of Saturday, 1,655 Kentuckians were hospitalized with COVID-19, according to Gov. Andy Beshear.