Rabbits are generally very cautious by nature, and when startled can leave a trail of droppings involuntarily. These are usually dry and easy (although understandably inconvenient) to clean up.
It may be the case, that since these new areas are "unfamiliar territory" your rabbit is possibly getting spooked when its exploring. If possible monitor the situation and see if it improves with time.
Alternatively, it may be attempting to mark its territory. While this is cute when they "chin" objectswhen they "chin" objects, droppingdroppings are less fun. However, again over time, when given the opportunity to feel at home in an area this behaviour should disappear.
Likewise, try and slowly introduce areas to your rabbit. Set up gates so they can only explore smaller areas and retreat to safety easily.