This paper addresses the Weighted First-Order Model Computation (WFOMC) problem, which computes the weighted sum of models of weighted first-order logic sentences. The bounds on the pieces for which WFOMC can be computed in polynomial time for the domain size lie between the two-variable piece ($\text{FO}^2$) and the three-variable piece ($\text{FO}^3$). While WFOMC for $\text{FO}^3$ is $\mathsf{P_1}$-hard, polynomial-time algorithms exist for $\text{FO}^2$ and $\text{C}^2$, although they can be extended to specific axioms such as the linear order axiom, the axiom of acyclicity, and the axiom of connectedness. Previous research has focused on extending the piece to axioms for a single distinct relation, leaving gaps in our understanding of the complexity bounds for axioms for multiple relations. This paper explores the extension of the two-variable piece to axioms for two relations, presenting both negative and positive results. We show that the WFOMC for $\text{FO}^2$ with two linear order relations and $\text{FO}^2$ with two acyclic relations is $\mathsf{ P_1}$-hard. Conversely, we provide a polynomial-time algorithm for the WFOMC of $\text{C}^2$ with a linear order relation, its successor, and another successor.