The holiday-themed series sees Renner's archer Avenger reluctantly taking on a protégé in Hailee Steinfeld's headstrong Kate Bishop.
"Hawkeye" follows the onscreen events of "Avengers: Endgame," during which Renner's grief-stricken Clint Barton took out his anger over Thanos snapping away his family by going on an underworld killing spree as the masked vigilante Ronin.
"I'm surprised I still fit the tights," Renner joked Monday on "GMA" about playing the character for a decade, first appearing in 2011's "Thor."
"I never could have imagined playing him this long," he continued, then adding that this series "ended up being a real good reason to keep continuing" his stint as Clint.
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After the Avengers and other MCU heroes saved Clint's family -- along with half of humanity -- Renner's character is trying to get back to a normal life as a dad.
"We start off with a very, very happy step forward into a family vacation into New York at Christmas and taking the kids to go see a musical, and then everything kind of goes sideways from there," Renner said at a recent press conference.
This is where Hawkeye mega-fan Kate comes in. "[S]he's inspired by him and motivated by him and wants to be at his level and is very overeager -- and he puts up with a lot," Steinfeld laughed.
Renner said Kate sees past Clint's "veneer of grumpiness," adding, "It comes from just the weight and the horrors and the tragedies and loss that come with the superhero game. And the lightness that Hailee's character brings in kind of counteracts that. It's really cathartic and, I think, quite beautiful."
As for what advice Renner gave Steinfeld about joining the MCU in "Hawkeye," he told "GMA" he told her "to stretch and to hydrate," noting that he was "proud" to see her come in "very prepared and always willing."
Steinfeld, who called her first foray in the MCU "surreal," told "GMA" on Wednesday it wasn't until she was in the thick of it that she realized he gave her some "prime advice. She said, "Stretch constantly [and] hyrdrate -- two things I definitely did throughout."
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Marvel Studios president Kevin Feige summed "Hawkeye" up: "Six episode, six days, will Clint make it home for Christmas? Which was fun and a breath of fresh air after world-ending stakes and Celestials bursting out of planets [in "Eternals"], and Multiverse shenanigans. This is, like Hawkeye himself, a grounded, family-based show."
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