Previously head coach of second-tier English side Featherstone, Carr joined the Dragons as an assistant this season but found himself thrust into the hot seat after the club axed Anthony Griffin last month.
The Dragons have won two of his four games in charge, most recently triumphing over undermanned South Sydney 36-30, and were gallant in a close loss to reigning premiers Penrith last week.
Saints were keen to have unveiled Griffin's full-time successor by now but had to go back to the drawing board when front-runner Jason Ryes knocked them back.
Premiership-winning ex-Cronulla mentor Shane Flanagan has now firmed as the favourite to take over, with fellow NRL assistants and former Dragons players Ben Hornby and Dean Young also in the frame.
Carr has been tight-lipped on his own head-coaching aspirations whenever quizzed during his month in charge.
The joint venture appears likely to name their new coach during their bye next week in the hopes the successful candidate can begin tinkering with the roster, but Carr is set to see the season out.
While head coaches often bring their own assistants with them to new clubs, the Dragons are hopeful there is room for Carr on the staff beyond 2023.
"He'd be a big loss if he was to go elsewhere, to be honest," prop Blake Lawrie told AAP.
"I've seen how he addresses the group, his knowledge. He's very motivated and he's direct with his messaging.Â
"I can't fault him."
Captain Ben Hunt believed Carr's positive influence had been reflected in an improved month on the field.
"Carry's doing a good job," the halfback said.
"He's got the locker room down there, the boys appreciate him. The last month, we've been playing some pretty good footy."
Amid speculation he could agitate for a release following the departure of confidant Griffin, Hunt confirmed reports Flanagan had contacted him this week.
During his time in charge, Griffin experimented with playing Hunt at hooker - the position his fills at State of Origin level - but Flanagan reassured Hunt he had no such plans.
"I had a small chat with Flanno," he said.
"It was just sort of about his intentions that he wanted to come in and coach and that if he did get the job, he wanted me to play halfback."
Lawrie urged the Dragons not to delay with their coaching call.
"Whatever decision they make, I hope they make it sooner rather than later so we can all buy into the direction that they want us to go," he said.
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