  In 1957, Hadwiger conjectured that every convex body in $\mathbb{R}^d$ can be
covered by $2^d$ translates of its interior. For over 60 years, the best known
bound was of the form $O(4^d \sqrt{d} \log d)$, but this was recently improved
by a factor of $e^{\Omega(\sqrt{d})}$ by Huang, Slomka, Tkocz and Vritsiou. In
this note we take another step towards Hadwiger's conjecture by deducing an
almost-exponential improvement from the recent breakthrough work of Chen,
Klartag and Lehec on Bourgain's slicing problem. More precisely, we prove that,
for any convex body $K \subset \mathbb{R}^d$, $$\exp\bigg( - \Omega\bigg(
\frac{d}{(\log d)^8} \bigg) \bigg) \cdot 4^d$$ translates of $\text{int}(K)$
suffice to cover $K$. We also show that a positive answer to Bourgain's slicing
problem would imply an exponential improvement for Hadwiger's conjecture.
