The guideline is that you can add fresh milk to frozen milk as long as the amount of frozen is greater than the amount of fresh you add. Otherwise you raise the temperature of the frozen milk into a risky range.
The usual strategy is to use breastmilk storage bags such as Lanisoh's, Medela's, Ameda's, or I am sure there are other options. Since breastmilk is good at room temperature for 8-10 hours and in the fridge for 5 days you can put your pump parts and a storage bag in the fridge for your daily pumping and then freeze those bags once the day is done.
Storing in smaller amounts additionally makes sense because once defrosted the thawed milk must be used within 24 hours rather than the usual 5 days that never-frozen milk can last in a fridge.