But it'll be worth sticking around for. After all, the champions have obvious needs at receiver, tight end and running back.
Whether they address those needs, of course, is uncertain. Reese, after all, believes in taking talent, regardless of position.
"You just wait and see if somebody is going to fall to you or if there is somebody you may be able to move up for, that you're in love with," Reese said. "It's always a good problem to have when you're picking 32."
The last time the Giants were at the bottom of the first-round pile, they cleaned up. They grabbed safety Kenny Phillips with the 31st pick overall in 2008, after New England forfeited its first-round choice because of "Spygate." And then, New York added cornerback Terrell Thomas in the second round, and receiver and recent Super Bowl hero Mario Manningham as well as linebackers Bryan Kehl and Jonathan Goff the next two rounds.
"We never try to reach in the draft," Reese said. "We're conscious of what we think our needs are, but we try to pick the best player in the draft. We do try to fill holes in free agency when we get an opportunity to do that, but we never say let's get a certain position in the draft.
"We just try to pick the best player."
The Giants have had a fair amount of success in the draft despite having only one Top 5 pick in the last eight years.
But champions or no, there are needs.
Manningham, whose spectacular sideline catch ignited the game-winning drive in the 21-17 win over the Patriots in the Super Bowl, signed with San Francisco as a free agent. Hakeem Nicks, the 29th choice in the 2009 draft, and free-agent super find Victor Cruz are back and Domenik Hixon is returning from knee surgery. But a good, young receiver would help the offense and this is a deep position in this draft.
New York also lost bruising running back Brandon Jacobs - the 110th pick in 2005- to San Francisco, so a backup for Ahmad Bradshaw is also needed.
If there is a desperate need, though, it might be at tight need. Starter Jake Ballard and backup Travis Beckum both suffered major knee injuries in the Super Bowl. Ballard does not expect to be ready until the middle of the season, while Beckum seemingly is ahead of schedule.
The Giants usually try to stockpile defensive linemen, where possible, and the difference in the Super Bowl was Manning and the defensive line. Pierre-Paul and Justin Tuck are cornerstones of a line that also features veteran Osi Umenyiora, who has a year left on his contract. Linval Joseph and Chris Canty are strong on the inside and the team still doesn't know what it has in Marvin Austin, the 2011 second-round choice who missed last season with a torn pectoral muscle.
Still, don't be surprised if the Giants take another pass-rushing lineman after losing veteran Dave Tollefson to free agency.