When the new force on an object increases, how does the object's acceleration change?

This is also contingent on the direction of the force. If the force is towards the motion of the body, then it will increase (speed up). If the force is acting against the direction of motion, then it will decrease (slow down). Think of a ball on the ground that is already rolling. Then you come up behind it and hit it with a stick. It goes faster. This also works in reverse for acting on in the opposite direction of an object in motion (slows it down).