First I define my version of the set cover problem: We have a collection of sets such as $S_1, \dots, S_m$ where each $S_i$ is a subset of $M=\{1,\dots, m\}$. The goal is to find the minimum number of $S_i$'s where their union is equal to $M$. This is my standard version of set cover.
Now, suppose each two consecutive set $S_i$ and $S_{i+1}$ in our problem differ in at most one item, i.e., $\big||S_i|-|S_{i-1}|\big| \leq 1$ and either $S_i\subseteq S_{i+1}$ or $S_i\subseteq S_{i+1}$. Does this assumption make the set cover easier to approximate (in polynomial-time)?