U.S. governors are pursuing different routes to restart local economies hit by the coronavirus pandemic, with several Southern leaders moving to reopen businesses while others voice concern about taking such steps without more robust testing capacity.

In Georgia, gyms, bowling alleys, barbers and other nonessential businesses were preparing to open as early as Friday, even as some mayors pushed back against the governor’s new order. Some retailers in South Carolina were already open, and affected businesses in most Tennessee...