Studies are inconclusive on whether it's okay to keep thawed breast milk which has been partially consumed. Once a baby begins drinking expressed human milk, some bacterial contamination occurs in the milk from the baby’s mouth. While some may advise you to dump your EBM (ouch) that is leftover from feeding, however based on related evidence thus far, it seems reasonable to discard the remaining milk within 1–2 hours after the baby is finished feeding instead of immediately. With regards to rerefrigerating vs keeping it out in room temp - no studies have been done, so we don’t know whether one method is preferred. We do know that for any food, refrigeration slows bacteria growth. If you are worried that the milk is spoilt, my suggestion is to do what you do with cow’s milk: Smell it and taste it!
This is a definite no-no! This is because once the milk is thawed to room temperature, the room temperature promotes bacteria growth in the breastmilk which may be harmful to your little one. What I did was to store my breastmilk is 80ml bags instead to minimise wastage instead of filling the bags up to maximum capacity. Sure, it may seem like wasting breastmilk bags, especially if you're using disposables but I was a low-supply mummy so every little bit of breastmilk that I could save, I would save. http://kellymom.com/hot-topics/frozen-milk-power-outage/
thanks