What you call an intersecting family is typically called an intersecting antichain. The value and realization of your unspecified $m_n$ is well known.
Taking all subsets of size $k$ gives an antichain which is intersecting provided $n\lt 2k.$
For $n=2t$ even, the largest antichain has size $m_{2t}=\binom{2t}t$ and consists of the sets of size $t.$ It is not intersecting. The family of $t+1$ elements sets is an intersecting antichain, in fact the largest one.
For $n=2t+1$ odd one has $m_{2t+1}=\binom{2t+1}t=\binom{2t+1}{t+1}$ realized by taking all the $t$ element sets or all the $t+1$ element sets. Since the second of these two maximum size antichains is an intersecting antichain, that answers the question and we see that here $m_n\gt m_{n-1}.$
Returning to $n=2t$ the intersecting antichain has mentioned has size $2m_{n-1}.$