Good idea for a survival tool (The reverse can be used as a hammer) but for killing zombies, you will want something designed for thrusting at the head or something which is designed to take off the head/limbs.
Henry rifles has both a .22 pump action and a few lever action models (henryrepeating.com) that are fairly compact and hold 12 to 15 rounds. If you have a dud round just pump it or use the lever action and you're back in business. A machete is fairly easy to get and wouldn't be a bad backup weapon either. Get a military surplus version if possible since they are made of higher quality steel than most other models.
Actually I was thinking of the plainer versions like the .22 lever action youth model. They're small and light enough to carry around all day and you can find .22 ammo pretty much anywhere ammo is sold. They also have that survival AR-7 model that breaks down and stores in it's own stock. Both the receiver and barrel on the AR-7 has a teflon finish for durability. I'd also want something a little bigger for trouble of the non-zombie variety (looters that thought my stuff was theirs for the taking).