This is not an answer to what the best known upper bound is, but rather a comment that the (known) average distribution of divisors indicates you might not expect to do any better than the bounds on the divisor function itself. Tennenbaum's "Introduction to Analytic and Probabilistic Number Theory", $\S$ 6.2 p. 207 says

For each integer $n$, let us define a
random variable $D_n$ taking the
values $\log d/\log n$, as $d$ runs
through the the set of the $\tau(n)$
divisors of $n$, with uniform
probability $1/\tau(n)$. The
distribution function $F_n$ of $D_n$
is then defined by $$
> F_n(u):=\text{Prob}(D_n\le
> u)=\frac{1}{\tau(n)}\sum_{d|n,d\le
> n^u}1\quad (0\le u\le 1). $$ It is
clear that the sequence
$\{F_n\}_{n=1}^\infty$ does not
converge pointwise on $[0,1]$.
However, we shall see the sequence of
Cesaro means $$
> G_N(u):=\frac{1}{N}\sum_{n\le N}F_n(u)
> $$ is *uniformly* convergent on $[0,1]$.
Remarkably, the limit is the
distribution function of a probability
law well known to specialists: the
arcsine law, with density
$1/(\pi\sqrt{u(1-u)})$. Large and
small values have high probability: if
$D$ is a random variable with this
distribution law, we have $$
> \text{Prob}(D<0.01\text{ or
> }D>0.99)\approx0.128 $$ This indicates
that, on average, an integer has many
small (and correspondingly many large)
divisors.

Update: One thing we learn is that the relevant parameter is not $N$, but $u:=\log_n(N)$.