For the conduction cooling of HTS magnets using coated conductor, the coils are either wet-wound or vacuum impregnated by cryogenic epoxy after dry winding. It is known that the magnet can be degraded after cool down due to delamination of the conductor. There have been several experimental investigations on the weak c-axis strength of the coated conductor. The structural delamination characteristics of the stacked conductor are also important for the design of mechanically stable conductively cooled HTS coil such as HTS generators, HTS motors, and SMES. C-axis tension experiments were performed with narrow soldering anvils to investigate the different delamination strengths at the edge and center of the conductor. FEM analysis was carried out to investigate the thermal stress caused by the different thermal expansion coefficients of several components of HTS coil for conduction cooling. The results of our analysis suggest possible coil configuration changes to avoid excess tensile stress at the edge of the conductor