This study examines the temperature-dependent evolution of the lattice constants for various CuAl$_2$-type compounds, including NiZr$_2$, (Co,Rh,Ir)Zr$_2$, (Fe,Co,Rh,Ir)Zr$_2$, and (Co,Ni,Cu,Rh,Ir)Zr$_2$, in the pursuit of negative or zero thermal expansion. Results reveal that NiZr$_2$ has positive thermal expansion, while the other compounds exhibit uniaxial negative thermal expansion along the $c$-axis contraction. The study suggests that the $c$-axis thermal expansion can be controlled by manipulating the $c/a$ ratio through $Tr$-site substitution, providing a design principle for achieving negative thermal expansion of the $c$-axis and potentially zero thermal expansion in a single compound in $Tr$Zr$_2$ compounds.