Hilton-Milner tells us about an intersecting family of $k$-size subsets $\mathcal{F}$ from $\binom{[n]}{k}$: If $|\mathcal{F}| > \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$, then all elements of $\mathcal{F}$ share at least 1 element of [n].

The theorem classifies the non-trivial $\mathcal{F}$ if equality occurs i.e. $|\mathcal{F}| = \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$.

I was wondering if there are known results/literature to see on classifications with $|\mathcal{F}| < \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$, particularly values very close to equality?

I am currently looking at the case of $n=7$, and $k=3$ to get an idea of this (i.e. what can be said about an intersecting family of size 12 < $\binom{6}{2}-\binom{3}{2}+1?)$.