In this paper, the authors briefly mention that one proposed method to bypass the Eastin-Knill theorem is to perform code-switching. That is, given codes $C_1$ and $C_2$ which permit a complementary set of transversal gates, one can encode their data using code $C_1$ and perform some logical operations there, and when one wants to implement a gate which is not transversal in $C_1$ but is transversal in $C_2$, switch to $C_2$ and implement the desired logical gate there. See here for details about how to implement code switching between the 5 and 7-qubit codes.
In principle, at least to me, this seems like it should work but they claim that "such schemes do not yield a set of universal operations". Does it not work in general, or is it that no one has yet found a pair of codes that do this?