Jobs Now Act of 2021 This bill directs the Department of Labor to carry out a two-year pilot program to award grants to general local governmental units or community-based organizations to retain, employ, or train employees who provide a local government unit with a public service (i.e., job creation). At least 50% of grant funds must be used to retain employees who would otherwise be laid off due to budget cuts. Labor must encourage grantees to use grant funds to retain, employ, or train veterans, individuals with disabilities, individuals receiving unemployment benefits, or dislocated workers.