The Rangers did not qualify for the playoffs for two seasons from 2017-19 after announcing they were going into a rebuild with a letter to fans Feb. 8, 2018.
"I don't know there's a set timeline where I'm going to say, or anyone can say, the rebuild is over," Drury said. "We're looking deep into everything in the organization to put ourselves in the best spot next October to get off to a great start and push toward making the playoffs."
Quinn said Wednesday he felt he and Drury have a lot in common.
"Chris and I have always had a great working relationship," Quinn said. "We've known each other a long time as well, and we certainly see a lot of the same things in the game of hockey. We've got a great deal of respect for each other. Not only do we have a great personal relationship, we've got a great working relationship."
Former Rangers president and general manager Glen Sather will work with Drury through the transition phase to help him get acclimated in his new roles.
Sather, an alternate governor and senior adviser to Rangers owner James Dolan, said he believes it can be easier for one person to be president and GM.
"You're more involved in everything, in every detail, in every decision," Sather said. "You are the guy that has to account for it and it's your responsibility to lead the team and to guide the team, and also take care of the scouting process with all the people that we have around the world scouting and trying to bring talent in.
"Chris is in a position where he's been involved in every part of the organization so far. He was a great player in the NHL and I think he's going to be terrific in this job."
Drury, who grew up a Rangers fan in Trumbull, Connecticut, played four seasons for New York from 2007-11 and was captain for the last three. He scored 615 points (255 goals, 360 assists) in 892 NHL games for the Colorado Avalanche, Calgary Flames, Buffalo Sabres and Rangers from 1998-2011 and 89 points (47 goals, 42 assists) in 135 playoff games. He won the Stanley Cup with the Avalanche in 2001.
Drury was hired by the Rangers as director of player development Sept. 6, 2015, promoted to assistant GM on Sept. 2, 2016, and to associate GM on Feb. 5.
The 44-year-old filled in as an assistant when Quinn and his staff were unavailable because of NHL COVID-19 protocol for six games from March 17-28.
"There's no doubt in my mind I know I'm ready, and in all honesty I owe that to Jeff Gorton," Drury said. "From the minute I got hired in development he put his arm around me and had me in every meeting whether it was with Mr. Dolan or amateur scouting meetings.
"You name it, I was everywhere, and I was fortunate enough to learn from him. Without question, I'm here today because of how he treated me and helped me these last six years."
Drury said part of what he has to do is go through a hiring process to fill the roles he previously had, including general manager of Hartford of the American Hockey League. He said he has started thinking about candidates.
"I think we do have a lot of key pieces, I think it's an exciting time for the organization," Drury said. "Just like every organization does when the season ends, we're going to take a look at everything do what we can to take the next step and make ourselves a playoff team.
"I obviously take this very, very, very seriously. My goal is to bring the Cup to [Madison Square Garden], period."
