METAIRIE, La. — With a 2-3 record, the New Orleans Saints certainly don’t like where they are as they go into their bye week.

But they do know it certainly beats the alternative going into a week of much-needed rest before the toughest stretch of their schedule begins with a road game against the Detroit Lions on Oct. 19.

They never want to look at it this way, but the Saints could easily be 1-4 after struggling in a 37-31 overtime victory over the Tampa Bay Bucs on Sunday. They were teetering on the brink of disaster when they fell behind by 11 points three times in the second half — twice in the fourth quarter.

The good news was that by the end of the crazy day, they were only one game behind the Carolina Panthers in the NFC South race with still plenty of time to make a run at the title in a weak division.

But there are lots of problems on both sides of the ball, to be sure.

Even though the Saints lead the NFL in total offense, a spot they’re accustomed to being in, and rank in the top seven in both passing and rushing, only three teams have turned the ball over more than their 10 giveaways.

On defense, the Saints have taken a step backward after being one of the league’s top units a year ago. They rank 25th in total yards allowed and are 26th against the pass and 14th against the run. They’re also 28th in scoring defense, giving up 28.2 points per game after ranking fourth a year ago at 19.0 per outing.

While the players got a chance to get some rest and get away from the team’s training facility for a week, particularly after several were injured against the Bucs, the coaching staff was searching for answers to their tough start.

Asked if he thought the first five games of the season would be so difficult, coach Sean Payton said he doesn’t go into a season anticipating anything.

“You really aren’t surprised by anything,” he said. “It is always challenging to win in our league. I say that honestly … it is hard. You learn about your team and you quickly go into making the corrections and trying to improve.”

As is his custom, Sean Payton hasn’t publicly talked about the status of All-Pro tight end Jimmy Graham since he left Sunday’s game with the Bucs after apparently spraining his right shoulder.

NFL.com reported Sunday night that Graham sprained the shoulder and would have further testing on Monday, but Payton, during his weekly news conference, said, “We’ll see where we are at.”

The Saints may have given a little bit of an indication on Tuesday, however, when they signed former Green Bay Packers and Bucs tight end Tom Crabtree after reportedly bringing in several players for a tryout.

The signing of Crabtree gives the Saints another body behind veteran Benjamin Watson and second-year pro Josh Hill if Graham isn’t able to practice, or, in a worst-case scenario, misses any games after their bye week.

To open a spot on the 53-man roster for Crabtree, the Saints put starting free safety Jairus Byrd on season-ending injured reserve Tuesday — four days after he underwent knee surgery.

Byrd, who suffered a torn lateral meniscus in a non-contact drill during practice on Oct. 2, was kept on the roster because he was hurt so late in the week and was listed as one of the team’s seven inactives for the game with the Bucs.

The loss of Byrd coincided with the return of safety Marcus Ball, a former Canadian Football League standout who was signed by the Saints during the off-season.

Ball was vying for a spot in the nickel and dime packages in the spring and summer but suffered a hamstring injury in the third exhibition game and missed the next five games.

Byrd’s spot in the starting lineup against the Bucs was taken by Rafael Bush, with Kenny Vaccaro taking his usual spot at strong safety.

The Saints also have rookie safety Vinnie Sunseri, although he has played mostly special teams in the first five games.