Question for working mamas that pump

How are you storing your breastmilk at work after you pump? I have a medela cooler bag with an ice pack that I've been putting in my work's mini fridge set aside in the room they have for pumping mamas. However, none of the other mamas have been using the mini fridge but me and it has made me paranoid that someone might mess with my baby's milk as the room seems to have people that do not pump in it constantly...

Would I be able to just keep the milk at my desk with the ice pack? Is putting it back in a fridge overkill?