This study presents a novel interpretation of complexity in the formalism of $f(R,{\mathcal{L}}_m,\mathcal{T})$ gravity, where R specifies the Ricci scalar, ${\mathcal{L}}_m$ is written as matter Lagrangian that describes characteristics of the matter fields, and $\mathcal{T}$ represents the trace of energy-momentum tensor. We commence with the contemplation of the inner sector as a cylindrical configuration having anisotropic matter and linked with the exterior region through a hypersurface. We determine the modified field equations associated with $f(R,{\mathcal{L}}_m,\mathcal{T})$ gravity and assume the preliminarily established relations between the conformal tensor and the intrinsic curvature. We consider the generally employed mathematical expressions for Tolman mass and C-Energy and evaluate their relationship with the conformal tensor. The structural scalars are calculated from orthogonal splitting of Riemann tensor, out of which YTF is categorized as complexity factor for the inferred fluid composition. It is important to specify that the complexity factor is eradicated in the framework of uniform matter configuration and also in the scenario when anisotropic terms are canceled with each other. Significant results related to the Weyl scalar, Tolman mass, and the complexity factor are derived in $f(R,{\mathcal{L}}_m,\mathcal{T})$ gravity. The vanishing complexity constraint is considered in order to deal with the modified field equations that have extra degrees of freedom and to find particular solutions corresponding to the specific models.