  Let $\epsilon > 0$ be sufficiently small and let $0 < \eta < 1/522$. We show
that if $X$ is large enough in terms of $\epsilon$ then for any squarefree
integer $q \leq X^{196/261-\epsilon}$ that is $X^{\eta}$-smooth one can obtain
an asymptotic formula with power-saving error term for the number of squarefree
integers in an arithmetic progression $a \pmod{q}$, with $(a,q) = 1$. In the
case of squarefree, smooth moduli this improves upon previous work of Nunes, in
which $196/261 = 0.75096...$ was replaced by $25/36 = 0.69\bar{4}$. This also
establishes a level of distribution for a positive density set of moduli that
improves upon a result of Hooley. We show more generally that one can break the
$X^{3/4}$-barrier for a density 1 set of $X^{\eta}$-smooth moduli $q$ (without
the squarefree condition). Our proof appeals to the $q$-analogue of the van der
Corput method of exponential sums, due to Heath-Brown, to reduce the task to
estimating correlations of certain Kloosterman-type complete exponential sums
modulo prime powers. In the prime case we obtain a power-saving bound via a
cohomological treatment of these complete sums, while in the higher prime power
case we establish savings of this kind using $p$-adic methods.
