The Saints and Hill agreed to a two-year deal, which will take the place of the one-year, $4.641 million tender they had previously offered him as a restricted free agent.
The deal is worth $21 million, including $16 million fully guaranteed at signing, plus $1 million more in performance incentives, a source told ESPN's Adam Schefter. Yahoo Sports was first to report the details of Hill's contract.
"I've felt a lot of love from the organization and the fan base. ... My wife and I always wanted to stay in New Orleans, so this worked out perfect," Hill said on a conference call Sunday.
Hill said the idea of a two-year deal "made the most sense for everybody" because it gives him a chance to become Drew Brees' successor if Brees retires in the next year or two -- but also gives him freedom if that doesn't happen.
"This gives me a clear trajectory and opportunity for me to compete and to be the guy in New Orleans in  if Drew decides to be done. And if he's not, then we'll figure out another contract or my role will continue to be the way it is," Hill said. "This was a way to prevent me to get to free agency next year, and it created some structure for both me and my family, as well as the organization."
In the meantime, Hill said he envisions playing the same kind of role he did in January's playoff loss to the Minnesota Vikings, in which he completed one pass for 50 yards, ran the ball four times for 50 yards and caught two passes for 25 yards and a touchdown.
Hill posted a message on Instagram on Sunday, saying, "I've had this dream since I was a little kid of playing in the NFL. The New Orleans Saints made that dream a reality. ... Thank you NOLA."
Hill, 29, did not sign an offer sheet with any other NFL teams before last week's deadline for restricted free agents, which gave the Saints exclusive negotiating rights. Depending on how they structure the new deal, they could create some extra salary-cap space to help make room for Winston.
Regardless of Winston's pending arrival, Saints coach Sean Payton has made it clear he believes the versatile Hill could potentially succeed Brees as New Orleans' starting quarterback.
As for the Saints' interest in Winston, Hill said he was not surprised or deterred.
"Coach and I had had a conversation before free agency hit, kind of what his plan was for me. I've had multiple conversations following that, and I knew that we were always gonna bring in another guy," Hill said. "I didn't know who it was gonna be. But no surprise there. I think Jameis is a good football player, and he's a good quarterback.
"Nothing has changed for the New Orleans Saints. The goal always has been, and I think will always be, to win the Super Bowl. And so I trust Coach Payton, he's created so many opportunities for me. And I think it's great."
But in the meantime, part of the reason the Saints are interested in adding a veteran QB such as Winston is so they can continue to use Hill in his vital role as a QB/RB/WR/TE/FB/special-teams standout.
"Taysom's earned this opportunity to be our 2 [at quarterback]," Payton told WWL Radio last month after former Saints backup Teddy Bridgewater signed with the rival Carolina Panthers. "But he also has earned the opportunity to play and help us win football games as a 1. And what I mean by that -- whether you call him a receiver, a tight end, a specialist, also a quarterback -- yeah, he's gonna play.
"He's too good a football player [not to be on the field]. He's one of our better football players."
Hill has attempted only 15 passes in his three-year NFL career -- including a 50-yard completion in New Orleans' 26-20 playoff loss to Minnesota in January. He is 7-of-15 passing for 169 yards with zero touchdowns and one interception.
But the 6-foot-2, 221-pounder has run for 410 yards and three touchdowns and has caught 25 passes for 265 yards and eight touchdowns in 42 games, including the playoffs.
Hill's best game was arguably his most recent one -- that playoff loss to the Vikings, in which he completed the 50-yard pass, ran the ball four times for 50 yards and caught two passes for 25 yards, including a touchdown.
Hill told Schefter earlier this offseason that he planned to consider his options in free agency but was "in no hurry to leave New Orleans."
"As I look at my career, I have goals and I have a vision for myself as to what I can be in the NFL," said Hill, who has always made it clear that he wants to be a starting NFL quarterback. "And there has been nothing that has been said, or I've never been treated in any way that would lead me to believe that the vision I have for myself is not the same vision that Coach Payton has for me, as well as the other guys on staff."
Hill got a late start to his NFL career because he went on a two-year church mission to Australia after high school, then spent five years at BYU, where he suffered a total of four season-ending injuries. But he was impressive enough as a dual-threat QB at BYU to finish his career with 6,929 passing yards, 43 TD passes, 2,815 rushing yards and 32 rushing TDs. He began his NFL career as an undrafted rookie with the Green Bay Packers in 2017 before the Saints claimed him off waivers on cut-down day.