Upper Body is Just Extra to Keep the Bar Close
You can counteract the bar bouncing forward with the upper body… to a degree.
There are limitations to it, but more importantly, remember that the more of your effort is putting horizontal force into the bar, the less is going into upward acceleration.
The solution to pushing the bar away isn’t doing more damage control with the upper body, but doing the lift better to prevent it.
The pull is your primary control of the bar path, and the pull under is secondary and additive—it’s not a substitute.
There will always be at least a little horizontal force at the top of the pull—it’s extremely difficult to avoid it completely unless you avoid contact, which is a bigger problem.
The key is doing everything you can to maximize the bar’s upward acceleration and minimize any slowing down from the poor mechanics resulting from excessive distance between the bar and body.
This does NOT mean being less explosive with the hips—it means directing that explosiveness correctly.
That’s done by being balanced, keeping the bar as close to the legs as possible before contact, and driving vertically with the legs as powerfully as you open the hips, and for the same duration.