Gray-coding through nested sets

We consider the following combinatorial question. Let $$S_0 \subset S_1 \subset S_2 \subset ...\subset S_m$$ be nested sets, where #$(S_i) = i$. A move consists of altering one of the sets $S_i$, $1 \le i \le m-1$, in a manner so that the nested condition still holds and #$(S_i)$ is still $i$. Our goal is to find a sequence of moves that exhausts through all subsets of $S_m$ (other than the initial sets $S_i$) with no repeats. We call this "Gray-coding through nested sets" because of the analogy with Frank Gray's theory of exhausting through integers while altering only one bit at a time. Our main result is an efficient algorithm that solves this problem. As a byproduct, we produce new families of cyclic Gray codes through binary $m$-bit integers.