Cook broke down in tears last week when he told teammates he was leaving the Rabbitohs to take up a two-year deal with St George Illawarra.
His exit comes after he was told Souths would not be offering him a new deal for 2026, and he was free to leave the club at the end of the season.
The 32-year-old's departure comes as the Rabbitohs plan for a second stint under serial premiership winner Wayne Bennett.
Bennett has already ruled out a play for Sydney Roosters dummy-half Brandon Smith.
And while Cook's exit will free up some salary cap space, the Rabbitohs veteran says the club does not need to go to market for a No.9.
"They won't get anyone else. (Mamouzelos) is the one," Cook said.
"I've been working alongside Pete for three years, but it feels like he's been here a lot longer.
"He knows the system, he knows the plays, and he'll do that job just fine.
"I've loved training alongside him, and I even get to play alongside him now. He's going to do the job fine, there'll be no issues there.
"He's the man for the club to back, and I'm sure they'll do that. There's no need to look elsewhere when it's right under your nose."
Despite his disappointment at leaving, Cook vowed to help prepare Mamouzelos to take over the No.9 jersey.
"No doubt me and him will work closely for the rest of the year," Cook said.
"Because one thing is for certain - you've got to leave this jersey in a better place. I'll continue to do that until I pass on the reins."
Cook has shown in the past month he can still play at a level close to his best, with his return to form coinciding with the Rabbitohs' mid-season resurgence.
He was their best in Saturday night's win over Manly, setting up two of their three tries in a 14-0 victory while also busting eight tackles.
Cook said his exit from Souths had come swiftly, in a matter of weeks going from believing he could earn an extension to being told he could move to the Dragons.
"There was a lot more that went into the decision than just me," he said.
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