We have grown $MgB_2$ superconducting thin films on the SiC buffer layers by means of hybrid physical-chemical vapor deposition (HPCVD) technique. Prior to that, SiC was first deposited on $Al_2O_3$ substrates at various temperatures from room temperature to $600^{circ}C$ by using the pulsed laser deposition (PLD) method in a vacuum atmosphere of ${sim}10^{-6}$ Torr pressure. All samples showed a high transition temperature of ~40 K. The grain boundaries of $MgB_2$ samples with SiC layer are greater in amount, compare to that of the pure $MgB_2$ samples. $MgB_2$ with SiC buffer layer samples show interesting change in the critical current density ($J_c$) values. Generally, at both 5 K and 20 K measurements, at lower magnetic field, all $MgB_2$ films deposited on SiC buffer layers have low $J_c$ values, but when they reach higher magnetic fields of nearly 3.5 Tesla, $J_c$ values are enhanced. $MgB_2$ film with SiC grown at $600^{circ}C$ has the highest $J_c$ enhancement at higher magnetic fields, while all SiC buffer layer samples exhibit higher $J_c$ values than that of the pure $MgB_2$ films. A change in the grain boundary morphologies of $MgB_2$ films due to SiC buffer layer seems to be responsible for $J_c$ enhancements at high magnetic fields.