  Let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free
graph on $n$ vertices which contains no independent sets larger than $s$. The
behaviour of $\mathrm{ex}(n,s)$ was first studied by Andr\'asfai, who
conjectured that for $s>n/3$ this function is determined by appropriately
chosen blow-ups of so called Andr\'asfai graphs. Moreover, he proved
$\mathrm{ex}(n, s)=n^2-4ns+5s^2$ for $s/n\in [2/5, 1/2]$ and in earlier work we
obtained $\mathrm{ex}(n, s)=3n^2-15ns+20s^2$ for $s/n\in [3/8, 2/5]$. Here we
make the next step in the quest to settle Andr\'asfai's conjecture by proving
$\mathrm{ex}(n, s)=6n^2-32ns+44s^2$ for $s/n\in [4/11, 3/8]$.
