I think paper towel rolls are your best option. Though, you can use straight cardboard to replace the paper towel inerards if you can't wait for them to be used and thrown out. With the roll, start out with the original diameter and use a small ring cut out to fit around the external edge of the front, and internal edge on the back, use another roll and cut down the side so it will fit inside the other; do the same thing with the ends as the prior. Repeat this 3 or 4 times. You should end up with something like this:
You can also make a paper towel roll substitute by crimping the length of one side of a straight cardboard piece until it is round enough. It would also be stronger then a normal roll.