Quite a lot is known beyond $GL_n$, but the results aren't always as clean and complete as for $GL_n$. This is (at least partly) because the Levi subgroups of more general reductive groups don't always look as "similar" to the full group as they do for $GL_n$ -- i.e. in $GL_n$ all Levi subgroups are products of smaller $GL_n$s, while in general you get Levi subgroups which are isomorphic to products of groups of various different types.

There are two things which you would need to generalize in order to get what you want. The first is a construction of a type for each block $R_s$ of the decomposition of the category $R(G)$ of smooth representations of $G$. Once you have a type $(J,\lambda)$ for $R_s$, you get an equivalence of categories between $R_s$ and modules over the $\lambda$-spherical Hecke-algebra $\mathcal{H}(G,\lambda)$. There are constructions of types for lots of $G$ and $s$ now: all blocks of $GL_n$ and its inner forms, all blocks of $SL_n$, and all blocks of classical groups (points in $GL_n(E)$ which are fixed by some Galois involution, for $E/F$ quadratic or trivial; essentially this means special orthogonal, symplectic or unitary) when the residue characteristic is odd. It seems like the construction of types for inner forms of classical groups isn't all that far away either, although it's certainly not yet complete. All of these constructions go through the same basic approach as the Bushnell--Kutzko theory; they're completely explicit and lead to types for which you'd hope the Hecke algebras are computable. There are probably a few other situations where something is known (spin groups, $G_2$), but I'm not so certain about what exactly we do know. There are also some results on the corresponding question for mod-$\ell$ representations of $GL_n$ and classical groups, but the relationship with Hecke algebras becomes much more complicated here.

The second is that, once you have a Hecke algebra $\mathcal{H}(G,\lambda)$ associated to a type, whether or not you can show that this is isomorphic to some tensor product of affine Hecke algebras, or something similar. Beyond $GL_n$, this is known for inner forms of $GL_n$, and for the blocks in classical groups which correspond to either the classical group itself, or a *maximal* proper Levi subgroup. I think there are some additional results in certain cases, such as for blocks defined on the Levi subgroup of a symplectic group which is of the form $GL_r^k\times Sp_{2n}$, but I'm not entirely sure which ones have been completely worked out. One notable omission from this list is $SL_n$; I don't think this has been written down anywhere, but it's understood since $SL_n$ is so nicely related to $GL_n$. The problem is that writing things down is a bit of a pain, because the Levi subgroups of $SL_n$ don't look very nice.

All of the above is via Bushnell--Kutzko theory. There are also completely general results on depth zero representations of arbitrary groups, which are quite a bit more straightforward.

The references for this are over a ton of papers, so instead of listing them I'll just point you to the various people in each case:

- $GL_n$ is due to Bushnell and Kutzko, as you know.
- Inner forms of $GL_n$ are due to Vincent Secherre and Shaun Stevens.
- $SL_n$ is Bushnell and Kutzko for the supercuspidal representations, and to David Goldberg and Alan Roche for the other blocks.
- Supercuspidal representations of classical groups are Shaun Stevens; types for the other blocks are Stevens and Michitaka Miyauchi. Stevens--Miyauchi also has the results on maximal Levis.
- Depth zero representations are due to Lawrence Morris.