A word w=w1w2⋯wn is alternating if either w1<w2>w3<w4>⋯ (when the word is up-down) or w1>w2<w3>w4<⋯ (when the word is down-up). The study of alternating words avoiding classical permutation patterns was initiated by the authors in Gao et al. [6], 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 Gao et al. [6].Further, we extend the enumerative results in Gao et al. [6] 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 132¯ and up-down words of odd length avoiding the consecutive pattern 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.