Probability mapping from lower to higher dimension with noise

Question

I'm aware that for a function $f : \mathbb{R}^m \to \mathbb{R}^n$ where $m < n$ and a random vector $z$ with probability $p(z)$ than $y = f(z)$ has a distribution of zero measure in $\mathbb{R}^n$ by Sard's Theorem. Note that although I'm aware of it that theorem is way above my knowledge to comprehend its proof. However, I'm interested in using something similar to the change of variable formula to find the density of $\hat{y} = y + \epsilon$, where $\epsilon \in \mathbb{R}^n$ and $\epsilon \sim \mathcal{N}(0,1)$. In this case, though the Jacobian is rectangular if we consider the combined vector $(x, \epsilon)$, so I'm not sure how to proceed there? A simple example would be $f(x) = Wx$.