For the case of $n\times n$ grid, I don't know the exact rate but I can provide an upper bound on the cover time. By the same Matthews method (Theorem 11.2 of the reference), the cover time is bounded from above by the (largest) hitting time multiplied by $2\log n$. The largest hitting time in this case is of order $n^2\log n$, which can be calculated through effective resistance and commute time identity (Propositions 9.16 and 10.6). Therefore, the cover time in the grid case can be bounded by $O(n^2(\log n)^2)$ too.

Actually, if you look at the notes on p. 152, for your original problem, the expected cover time $E(\tau_{cov}) \sim \frac4\pi n^2(\log n)^2$.