You could use the chainlink for the upper half of the whole run..
maybe add a few boards of wood to the bottom half. And use the hardwire to make a skirt around the entire run.
(skirt meaning have it come up the side of the walls a foot and bury it a few inches and out...think L shape)
Chicken wire will not keep predators out.
Could you post a picture of this run area?
You may be able to add the hardwire around the bottom...up a few feet.