This study investigates and compares dynamic hedging strategies for American put options in financial markets governed by jump diffusion dynamics. Since closed-form continuous-time hedging solutions are generally unavailable in this setting, we approximate the underlying continuous-time models by using a discrete-time lattice framework. By focusing on the challenges posed by continuous-time hedging, we utilize a discrete-time and discrete-outcome stochastic framework with backward dynamic programming to optimize hedging under jump risk. The strategies include global quadratic hedging, local risk minimization, and Delta-Gamma hedging. We also examine an internal exercise specification in which the exercise decision is linked to the current hedging portfolio value, and compare it with the standard externally imposed exercise rule. Through numerical experiments, we compare the hedging error distributions across these methodologies, particularly highlighting how the inclusion of jump risk affects hedging outcomes. Moreover, we investigate and compare the Kou and Merton jump-diffusion models calibrated to S$\&$P 500 and Bitcoin (BTC) data. Our results indicate that while all strategies effectively manage standard market risks, local risk minimization offers superior performance in extreme market conditions. Additionally, incorporating a third hedging asset does not lead to a consistent improvement in the performance of the global hedging strategy. Although the additional option enlarges the hedge set and may reduce the required initial investment, it does not deliver a robust reduction in tail-risk measures under the jump diffusion setting considered here.