  A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1<w_2>w_3<w_4>\cdots$
(when the word is up-down) or $w_1>w_2<w_3>w_4<\cdots$ (when the word is
down-up). The study of alternating words avoiding classical permutation
patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it
was shown that 123-avoiding up-down words of even length are counted by the
Narayana numbers.
  However, not much was understood on the structure of 123-avoiding up-down
words. In this paper, we fill in this gap by introducing the notion of a
cut-pair that allows us to subdivide the set of words in question into
equivalence classes. We provide a combinatorial argument to show that the
number of equivalence classes is given by the Catalan numbers, which induces an
alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}.
  Further, we extend the enumerative results in~\cite{GKZ} to the case of
alternating words avoiding a vincular pattern of length 3. We show that it is
sufficient to enumerate up-down words of even length avoiding the consecutive
pattern $\underline{132}$ and up-down words of odd length avoiding the
consecutive pattern $\underline{312}$ to answer all of our enumerative
questions. The former of the two key cases is enumerated by the Stirling
numbers of the second kind.
