The correct exponent is $d=m-n+1$. Your question is essentially the same thing as: consider a matrix with uniform i.i.d. entries taking values in $[-1,1]$ (this collection of matrices has volume that is a constant factor greater than the ones with norm at most 1); then for what proportion of such matrices is the $n$th row within $r$ of the span of the previous rows?
I am using a claim here: the $n$th singular value of an $n\times m$ matrix agrees up to a bounded factor with $\min_{i}d(r^{(i)},\mathop{lin}\{r^{(j)}\colon j\ne i\})$ that I put in a preprint that I'm just finishing.
So: given the first $n-1$ rows, you're asking what is the probability that the $n$th row lies within $r$ of their span. The span is an $(n-1)$-dimensional space and an $r$-neighbourhood of such a space has volume of the order of $r^{m-(n-1)}$.