"Unforgettable" is a well-made cop show on CBS that could have easily been a distant memory by now. Instead, it begins the second half of its second season on Friday night, a case study in how to give more chances to a decent show that didn’t find its natural home immediately.

The series stars Poppy Montgomery as Carrie Wells, a police investigator in New York City with a blessing-and-curse gift: She has total-recall memory. She can, for instance, replay a crime scene in her head days later to spot a vital clue that didn’t seem important at first.

Montgomery pairs well with Dylan Walsh as Al Burns, a detective who, years earlier in Syracuse, N.Y., was her boyfriend. Al knows about Carrie’s gift and also about a ghost from her past. When Carrie was a girl, her sister was murdered, a crime that she witnessed, but that her otherwise supermemory refuses to recall.

The show made its debut in the fall of 2011, and the premiere drew a solid audience: almost 14 million viewers, according to the website TV by the Numbers. It went on to average about 12 million, but when the season ended the following spring, CBS announced that the show was being canceled. Then, a few weeks later, came an unusual announcement: The cancellation was being canceled. "Unforgettable" would become a summer series.

"When I heard it was uncanceled, I thought it was a joke, because that never happens," Montgomery, a television veteran whose resume includes a long run on "Without a Trace," said by phone. "I was pretty flabbergasted. It takes a lot to throw me, because I’ve seen so much in TV."

Ed Redlich, who created the show with John Bellucci, said the reprieve was the second time the series had shown it had a charmed life. The first was in its pilot-season phase: It had actually been pitched for the year before, but getting the right cast proved difficult.

"We didn’t get on the schedule, and it got to the point where everyone except my 12-year-old son thought we weren’t ever going to get on," he recalled.

"Unforgettable" has done more than simply hang on. It has evolved as a series to try to find a sustainable premise and tone. In the initial episode, Carrie had left police work; she was merely a witness on a case Al was investigating. But later, she joined him as a New York police officer in Queens.

And at the start of Season 2 last July, their base of operations moved to Manhattan, a shift that gave the writers a chance to graduate from street crimes to the more glamorous world of the rich and powerful.

Carrie has also evolved; she broods less over her sister’s murder.

"In Season 2, she’s kind of decided to move on and not be stuck in the pain of her past," Montgomery said. "And so you see her actually get to be enjoying this gift that she has. I said to my mom, ‘All that dental work has paid off, because you’re actually going to get to see my smile in Season 2.’"

That has let the writers inject more humor into the show, and the relationship between Al and Carrie proves satisfying. It’s strictly professional — or is it? Friday’s episode opens with a nice wink, seeming to suggest that the two have married during the midseason layoff. The lighter tone fits the summer slot better; it’s a carefree time of year.

Lots of promising shows die prematurely, and, these days, needlessly. Networks have options: Change the time or the season, or even, for those that own multiple properties, the channel. So do the studios that make shows: Witness "Cougar Town," which was on ABC but is now on TBS. It would be nice to see more use of that flexibility. There are too many bad shows on TV to let decent ones wither.

Neil Genzlinger, New York Times