Can I add fresh milk to previously pumped milk that's been refrigerated?

I've been using the Hakkaa to catch my let down while my daughter feeds on the opposite breast. I've done it twice now and get about an ounce per feeding. Even though I put that in a bottle she still seems hungry afterwards. I'm also giving her formula after nursing/Hakkaa since I don't know how much she's actually getting from me.

I was thinking in order to start building my supply, to let her nurse and give formula afterwards, but then refrigerate whatever I get from the Hakkaa.