In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend yield $y$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as 

$$
\Delta\equiv \frac{\partial V}{\partial S_t}=e^{-y\tau}\mathrm{N}\left(d_1\right)
$$
with 
$$
d_1\equiv \frac{\ln S- \ln X +(r-y+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}}
$$

Let $B\equiv Xe^{-r\tau}$ the discounted strike and $\tilde{S}\equiv Se^{-y\tau}$ the 'yield-discounted' spot price, then 

$$
\begin{align}
\frac{\partial \Delta}{\partial \sigma}&=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\
&=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln \tilde{S} - \ln B }{\sigma^2\sqrt{\tau}}\right)
\end{align}
$$

As $\mathrm{n}\left(d_1\right)> 0$ whenever $\sigma\sqrt{\tau}>0$, the sign of the change in $\Delta$ as a function of $\sigma$ depends on whether

$$
\frac{1}{2}\sigma^2\tau \lessgtr\ln \tilde{S} - \ln B
$$
i.e. whether the (logarithmic) moneyness is within 1/2 of the term variance. HTH?