If you wake up to a Zombie Apocalypse, your primary concerns are going to be finding help and getting information. You're not going to just try one random internal phone in a hospital and give up. You're going to look for other phones, for cell phones, for Blackberries, you're going to turn on TVs and radios, you're going to look for cars with XM receivers, you're going to try the Internet at the hospital, at home, at the Police Station. Like Mr. Adventure says, you'd have no idea how widespread the phenomenon is. And most people, rather than assuming that nothing works, would continue to try these things over and over no matter how many times they fail to work. It would have been to the story's advantage to show this, demonstrating vividly how serious the situation is, as well as the character's growing fear and frustration. As it is, I don't remember there being any evidence that this world even has Internet or cell phones or XM.
There seemed to be a fair amount of time compression in the scenes between him waking up and him leaving the hospital; he was a lot surer on his feet when he found the exit. He may have tried numerous computers and radios and TVs and cell phones, I don't think the viewer needs to be shown all of that. As for the storytelling, I think it was more effective how it was done, leaving the audience in the dark until the situation could be dramatically related first-hand by the survivor. Those retelling scenes were some of the best, IMO.