In our previous report, we demonstrated the growth of high-quality epitaxial Co-doped ${\rm Ba}{({\rm Fe}_{1-{\rm x}}{\rm Co}_{\rm x})}_{2}{\rm As}_{2}$ (Ba-122) thin films using single crystalline ${\rm SrTiO}_{3}$ (STO) templates on $({\rm La},{\rm Sr})({\rm Al},{\rm Ta}){\rm O}_{3}$ (LSAT) substrates with pulsed laser deposition (PLD). Here, we report the dependence of the structural and superconducting properties of Ba-122 thin films on STO templates that were 2-150 unit cell (u.c.) thick. We obtained genuine epitaxial thin films with high crystalline quality and excellent superconducting properties for templates above a critical thickness, which was 50 u.c. for our growth condition. Although the quality of Ba-122 is excellent above the critical thickness, the best crystalline quality and superconducting properties were obtained on 100 u.c.-thick STO templates. The best-quality Ba-122 thin films show an on-set transition temperature $(T_{c,\ on-set})$ as high as 22.8 K, a transition width $(\Delta T_{c})$ as narrow as 1.3 K, and critical current density $(J_{c})$ as high as 3 ${\rm MA/cm}^{2}$ .