  A classical result by Hajnal and Szemer\'edi from 1970 determines the minimal
degree conditions necessary to guarantee for a graph to contain a $K_r$-factor.
Namely, any graph on $n$ vertices, with minimum degree $\delta(G) \ge
\left(1-\frac{1}{r}\right) n$ and $r$ dividing $n$ has a $K_r$-factor. This
result is tight but the extremal examples are unique in that they all have a
large independent set which is the bottleneck. Nenadov and Pehova showed that
by requiring a sub-linear independence number the minimum degree condition in
the Hajnal-Szemer\'edi theorem can be improved. We show that, with the same
minimum degree and sub-linear independence number, we can find a clique-factor
with double the clique size. More formally, we show for every $r\in \mathbb{N}$
and constant $\mu>0$ there is a positive constant $\gamma$ such that every
graph $G$ on $n$ vertices with $\delta(G)\ge \left(1-\frac{2}{r}+\mu\right)n$
and $\alpha(G) < \gamma n$ has a $K_r$-factor. We also give examples showing
the minimum degree condition is asymptotically best possible.
