The Next Mission Act directs the Secretary of Defense to establish a grant program that provides supplemental career services to military members and their spouses transitioning to civilian life. Grants will go to nonprofit organizations that offer services such as career assessments, job interview preparation, industry-specific training pathways, and job placement assistance to eligible individuals at no cost. To qualify for grants, organizations must be private nonprofits with a proven track record in career services and job training, demonstrate rigorous evaluation practices, have partnerships with employers and other entities, and be able to provide online training to reach rural participants. The program must be established within one year of enactment and will operate for three years, after which the Secretary of Defense must submit a report to Congress on outcomes including the number of individuals trained and employed, job retention rates, and median salaries. The bill also requires grantees to assess whether the program could be expanded to reserve military members and service members located overseas.