The concept of almost contact structures [\textit{D. E. Blair}, Contact manifolds in Riemann geometry. Lect. Notes Math. 509, Springer-Verlag (1976; Zbl 0319.53026)] has led to an array of related structures on odd-dimensional manifolds, such as Sasakian and cosymplectic structures. \textit{I. Vaisman} [Lect. Notes Math. 792, 435-443 (1980; Zbl 0431.53030)] introduced the geometry of locally conformal cosymplectic spaces. Later on, many other geometers contributed to this theory. The present paper is devoted to the interplay between Weyl geometry and locally conformal cosymplectic structures. In particular, it provides the definition of the almost contact Weyl structure and proves that in dimension \(\geq 5\) this structure leads to locally conformal cosymplectic structure. Additionally, the author utilizes the so-called Gauduchon metric [\textit{P. Gauduchon}, J. Reine Angew. Math. 469, 1-50 (1995; Zbl 0858.53039)] to characterize compact locally conformal cosymplectic Einstein-Weyl manifolds.