We adapt an argument of Tao and Vu to show that if $λ_1\le\cdots\leλ_d$ are the successive minima of an origin-symmetric convex body $K$ with respect to some lattice $Λ<\mathbb{R}^d$, and if we set $k=\max\{j:λ_j\le1\}$, then $K$ contains at most $2^k(1+\frac{λ_k}2)^k/λ_1\cdotsλ_k$ lattice points. This provides improved bounds in a conjecture of Betke, Henk and Wills (1993), and verifies that conjecture asymptotically as $λ_k\to0$. We also obtain a similar result without the symmetry assumption.