When adding angular momentum S and L to get J, J can range from absolute value of (L-S) to L+S in integer steps. Then once you have J you can figure the allowed values of m_j just like you do for S and m_s

When adding angular momentum $S$ and $L$ to get $J$, $J$ can range from absolute value of $|L-S|$ to $L+S$ in integer steps. Then once you have $J$ you can figure the allowed values of $m_j$ just like you do for $S$ and $m_s$