With these points in mind, I would highly recommend you consider looking into Road Show. It's a Lawlor game, so plenty deep enough, has varied and unique shots that are integrated into modes very well and is probably as "country themed" as you're gonna get. Road show was done one year after Twilight Zone, so some of it's rules are inspired by Lawlor's previous games. The only downside is linearity (modes cannot be stacked and must be done in order to an extent), but as long as you have something else to play (HS2) you'll be fine. Bowen Kerins recently did a tutorial and goes into detail about the gameplay here:
Good luck with the hunt buddy!