Solid Start Act of 2021 This bill permanently authorizes and expands the Solid Start program, which is an outreach program for veterans in their first year of separation from the military. Specifically, the bill requires the Department of Veterans Affairs (VA) to coordinate with the Department of Defense to collect updated contact information during transition classes or separation counseling for members of the Armed Forces who are separating from service; explain the existence and purpose of the program; call veterans three times within the first year of separation; publish information about the program in booklets and on the VA website; provide women veterans with information tailored to their health care and benefit needs; provide information on access to state and local resources, such as Vet Centers; gather and analyze data that assesses the effectiveness of the program; ensure calls are tailored to each veteran's needs by conducting quality assurance tests; and prioritize outreach to veterans who have accessed mental health resources prior to separation from the Armed Forces. Under the bill, the VA is authorized to (1) encourage members of the Armed Forces to provide additional contact information if they are unreachable following their separation, (2) follow up on missed phone calls, and (3) reach out to veterans who separated prior to the initiation of the Solid Start program.