  Let $p=\frac{1+\varepsilon}{n}$. It is known that if
$N=\varepsilon^3n\to\infty$ then w.h.p. $G_{n,p}$ has a unique giant largest
component. We show that if in addition, $\varepsilon=\varepsilon(n)\to 0$ then
w.h.p. the cover time of $G_{n,p}$ is asymptotic to $n\log^2N$; previously
Barlow, Ding, Nachmias and Peres had shown this up to constant multiplicative
factors.
