Uniform intersecting families with large covering number

A family $F$ has covering number $\tau$ if the size of the smallest set intersecting all sets from $F$ is equal to $\tau$. Let $M(n,k,\tau )$ stand for the size of the largest intersecting family $F$ of $k$-element subsets of $\{1,\dots ,n\}$ with covering number $\tau$. It is a classical result of Erdős and Lovász that $M(n,k,k)\le k^k$ for any $n$. In this short note, we explore the behavior of $M(n,k,\tau )$ for $n<k^2$ and large $\tau$. The results are quite surprising: For example, we show that $M(n,k,\tau )\left(\begin{array}{cc}=(1-o\left(1\right))\left(\frac{n-1}{k-1}\right),& \text{if }n=\lfloor k^{3/2}\rfloor \text{ and }\\ & \tau \le k-k^{3/4+o\left(1\right)}\\ & \text{ as }k\to \infty \\ <e^{-c{k}^{1/2}}\left(\frac{n}{k}\right),& \text{if }n=\lfloor k^{3/2}\rfloor \text{ and }\\ & \tau >k-\frac{1}{2}{k}^{1/2}.\end{array}\right)$