I’m using calls to CVX within a larger coordinate descent method on a convex function (it’s too big to do in one piece) and thus have an initial guess for the variables in my optimization left from the last iteration (other than the ones CVX computes automatically and doesn’t leave behind, I guess).

Hopefully this isn’t an obvious/easy/already-explained-somewhere question, but: Is there a way I can provide my old solution as an initial guess for the next call?

No, there is not. This is not as straightforward as it may seem. For one thing, CVX generates quite a few internal variables when converting your problem to solvable form. You would have to supply initial values for all of those variables as well to have hope to provide a “good” solution. Furthermore, research on warm-start methods for interior-point problems is actually somewhat immature. It is by no means clear that supplying the solution from a previous problem would improve performance significantly. In fact, it can degrade performance in some cases, because the starting point is near the boundary of the feasible set.