There are a number of sources of high-dimensional lattices that admit efficient CVP algorithms (and a few that admit efficient BDD algorithms, a "promise" version of CVP where one decodes errors contained in $\mathcal{B}_n(\lambda_1(L) / 2)$, e.g. the largest circumscribed sphere within the Voronoi cell of the lattice).
I'll briefly summarize some below, before discussing density.
CVP:
If one can efficiently solve CVP on $L$, then one can efficiently solve it on $L\otimes \mathbb{Z}^n \cong \bigoplus_{i = 1}^n L$. It sounds like you have already realized this, but it is worth mentioning explicitly.
The lattices $A_n, A_n^*, D_n, D_n^*$ can all be efficiently decoded (this is mentioned in Conway and Sloane)
A mild generalization of the $A_n, A_n^*$ lattices known as Coxeter lattices admit $O(n)$ CVP algorithms.
BDD:
Barnes-Wall Lattices admit $O(n(\log n)^2)$-time decoding algorithms
A particularly dense family of lattices are the Discrete Logarithm lattices. The precise complexity of BDD on it depends on a number of choices one makes in constructing it, but looks like it can be $o(n^2)$ pretty easily.
There are other dense high-dimensional lattices that admit fairly efficient BDD algorithms (namely Barnes-Sloane lattices, which one can even list decode), but I think these are $\Omega(n^2)$ (but poly-time), although the algorithms are fairly naive, so with further work it is plausible they can be decoded in $o(n^2)$ time.
Of all of these, I believe the densest are the Barnes-Sloane lattices and the Discrete Logarithm lattices, so if you can tolerate a BDD algorithm instead of a CVP algorithm, I would look into the discrete logarithm lattices. If you require CVP algorithms, you should probably look into Coxeter lattices, although I do not know how their density will compare with using direct sums of the leech lattice.
