Growth spurt / clustering - giving little extra?

My son is currently 4 weeks old and the last 3 days he wants to drink all the time in the evening, while there is no milk left. He keeps sucking, everything gets sore but nothing comes out. Then he cries until he gets more milk. It's really frustrating for both of us. I'm very tempted to give him some of the pumped milk stash, but I'm worried that this might mess with the supply&demand of milk in the long run. Does anyone recognize this? How do you deal with it?

At other times during the day & night there's more than enough milk.