The paper The Relationship between Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms contains some results of interest, although they are somewhat technical.
Specifically, it needs:
Smoothness Assumption: For $n\in\mathbb{N}$, define
$\nu(n)$ to be the minimum, over $d\in [n-2\sqrt{n}+1, n+2\sqrt{n}+1]$ of the largest prime factor of $d$.
The smoothness assumption is that $\nu(n) = (\log n)^{O(1)}$.
In this setting, if one has some small "advice string" specific to $G$ (the paper states one needs the large prime factors of $|G|$ and certain parameters of elliptic curves --- the total advice is of length $O(\log |G|)$, then:
Corollary 5. If the smoothness assumption is true, then there exists a polynomial-time generic (non-uniform) algorithm computing discrete logarithms in cyclic groups of order $n$, making calls to a DH oracle for the same group, if and only if all the multiple prime factors of $n$ are of order $(\log n)^{O(1)}$.
Here, "multiple prime factors" mean mean prime powers $p^e \mid n$ for $e > 1$.
If all prime factors of $n$ are "single" (e.g. $n$ is square-free), it appears they can do somewhat better --- their theorem 2 covers this case, and appears to remove the requirement of knowledge of the elliptic curves + the smoothness assumption (one still needs the factorization), and they explicitly evaluate the complexity of the reduction. I won't copy it here, as the theorem statement is somewhat long.
This is all to say that under a certain number-theoretic assumption, in the non-uniform setting there is no gap between DLOG and CDH.
