  We introduce the concept of \emph{pre-Jaffard family}, a generalization of
Jaffard families obtained by substituting the locally finite hypothesis with a
much weaker compactness hypothesis. From any such family, we construct a
sequence of overrings of the starting domain that allows to decompose stable
semistar operations and singular length functions in more cases than what is
allowed by Jaffard families. We also apply the concept to one-dimensional
domains, unifying the treatment of sharp and dull degree of a Pr\"ufer domain.
