Final answer:
The factor most likely to increase the likelihood of African American clients continuing with counseling is the counselor's willingness to 'go the extra mile' in understanding and addressing the cultural and individual needs of their clients.
Step-by-step explanation:
The question asks about the factor that increases the likelihood of African American clients continuing with counseling. While the provided reference doesn't give the answer directly, research in counseling psychology suggests that a counselor's willingness to be culturally competent and attuned to the specific needs of their clients greatly impacts the therapeutic relationship and client retention. This includes a counselor's dedication and ability to 'go the extra mile' in understanding the cultural, societal, and individual aspects of their clients' lives.