Has the derandomization of slightly non-uniform classes, e.g BPP/linear, been studied?
I used the this scholar search, and the only result which seems somehow relevant is this:

Lance Fortnow, Adam Klivans, Linear Advice for Randomized Logarithmic Space, ECCC TR05-042, 2005.

Unfortunately, I think it is not exactly what you need. In particular, the main result of the paper is $\mathbf{RL} \subseteq \mathbf{L}/O(n)$. Yet I believe the techniques used there might be helpful in the case of your question.