A recent characterization of all lengths of Plotkin-optimal quaternary (that is, over the ring $\mathbb{Z}_4$) codes of arbitrary type \cite{tang2025plotkin} also pins down the parameters for which no Plotkin-optimal code exists. In this article, we determine the best achievable parameters in several of these cases, obtaining codes whose minimum Lee distance is one less than the Plotkin bound, namely the codes with Plotkin-defect 1. To the best of our knowledge, this is the first attempt to study quaternary codes with Plotkin-defects. Precisely, we construct infinite families of quaternary $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee weight distributions. As a result, we find two quaternary linear code families with Plotkin-defect 1 and report at least 23 new or improved parameters having small (upto 4) Plotkin-defects, including 13 projective and 7 optimal parameters. We additionally report 4 quaternary linear codes with best-known parameters that are also projective. Further, their linear Gray images give two infinite families of distance-optimal, one infinite family of at least almost dimension-optimal binary linear codes and five infinite families of minimal binary linear codes.

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