One related structure is an incidence geometry. Each pair of points determine a unique line. Every $2$-design with $\lambda = 1$ is an example of an incidence geometry, but incidence geometries are more flexible.

Given an incidence geometry on $V$, you can induce an incidence geometry on $W \subset V$ by taking the lines to be the intersections of lines with $W$ (except you throw out the intersections of size at most $1$). This should let you construct incidence geometries whose parameters fit your problem out of block designs on slightly larger sets.

One related structure is an incidence geometry. Each pair of points determine a unique line. Every $2$-design with $\lambda = 1$ is an example of an incidence geometry, but incidence geometries are more flexible.

Given an incidence geometry on $V$, you can induce an incidence geometry on $W \subset V$ by taking the lines to be the intersections of lines with $W$ (except you throw out the intersections of size at most $1$). For example, there are Steiner triple systems (block designs where the blocks have size $3$ and $\lambda=1$) if and only if the number of vertices is $1$ or $3$ mod $6$. There is no Steiner triple system on $11$ vertices, but you can take a Steiner triple system on $13$ vertices with $26$ blocks/lines and delete $2$ points (and the line through them which only contains $1$ point now) to get an incidence geometry on $11$ points with $25$ lines so that each line contains $2$ or $3$ points.

This might not work well over all sets of parameters. Block designs with more than $1$ block have more blocks than points (Fisher's inequality). The same is true for incidence geometries (de Bruijn–Erdős). Projective planes are examples of equality. If you have fewer workers than files, so that workers have to handle more than about the square root of the number of files, then you aren't looking for a small deviation from a block design.