  A totally ordered monoid, or tomonoid for short, is a monoid endowed with a
compatible total order. We deal in this paper with tomonoids that are finite
and negative, where negativity means that the monoidal identity is the top
element. Examples can be found, for instance, in the context of finite-valued
fuzzy logic.
  By a Rees coextension of a negative tomonoid $S$, we mean a negative tomonoid
$T$ such that a Rees quotient of $T$ is isomorphic to $S$. We characterise the
set of all those Rees coextensions of a finite, negative tomonoid that are by
one element larger. We thereby define a method of generating all such tomonoids
in a stepwise fashion. Our description relies on the level-set representation
of tomonoids, which allows us to identify the structures in question with
partitions of a certain type.
