I keep cash in the compartment that's "in the flap". You do have to fold your bills, but for me that's not a problem since I rarely carry cash and close to never do I have more than two bills and a few coins. Right now I think there's one bill and 4-5 coins and that's it.Can I ask where you're keeping the bills? Do you have 2 to fold them?
I love using this piece for travel as well, I just empty out everything but my credit/debit cards, SI&TI cards and ID. If I have to carry two types of currency I usually just use one of the center compartments for the currency I need that day and the other gets put in the back where I keep my money now.