Combine the peach jam, olive oil, soy sauce, mustard, garlic and jalapeno in a medium bowl and season with salt and pepper. Reserve 1/2 cup.
Preheat the grill. Brush the chicken with olive oil and season with salt and pepper. Place the chicken skin-side down and cook until golden brown, 6 to 7 minutes. Turn over and continue cooking for 5 to 6 minutes. Brush both sides with the peach glaze and continue cooking until done, an additional 4 to 5 minutes.
Place the peach halves cut side down on the grill and grill for 2 minutes. Turn over, brush with the reserved 1/2 cup of peach glaze and grill until the peaches are soft, 3 to 4 more minutes.