We analyze American put options in a hyper-exponential jump-diffusion model. Our contribution is threefold. Firstly, by following a maturity randomization approach, we solve the partial integro-differential equation and obtain a tight lower bound for the American option price. Secondly, our method allows us to disentangle the contributions of jump and diffusion for the American early exercise premium. Finally, using American-style options on S\&P 100 index from 2007 until 2013, we estimate a range of hyper-exponential specifications and investigate the implications for option pricing and jump-diffusion disentanglement. We find that jump risk accounts for a large part of early exercise premium.