Does Tru sleep? I've made a study- okay, I re-watched the episodes and looked at the clocks. She must divide her sleep on a normal day. Here's why.
In the pilot, she was at Rebecca's apartment at 10 pm, but at the Morgue at 10:55 pm, when the cancer patient came in. So, her shift must start at 10:30 pm.
Her rewound days always (all three times) start at 8 am, so she must normally be home and in bed by then, otherwise she would rewind to a different location, but the same time.
So, if her shift is eight hours, it ends at 6:30 am and she would be in bed at the earliest at 7 am. If she has a shorter shift (six hours), she would still be in bed at 5 am at the earliest.
In Putting Out the Fires, she had brunch with Harrison at about 10:30 am. (Harrison being a night owl too, he would not have made Tru get up earlier than normal to have brunch with him.) Therefore, she slept, at most, five hours between 5 am and 10 am.
So, I'm betting that on a regular day, she has to nap in the afternoon.