Maybe they should call it flee agency. Every player the Giants seem to really want goes running back to his old team.

The official signing period for NFL free agents does not begin until 4 p.m. today, but the Giants already have missed out on two players for whom they planned to make significant pushes. Safety Devin McCourty re-signed with the Patriots on Sunday night and defensive end Brandon Graham re-signed with the Eagles on Monday. Both were high on the Giants' wish list and the team had been in contact with their camps during the "legal tampering" period that began Saturday.

The primary targets may be gone, but the needs are not. With those players unavailable, the Giants will have to look to a second tier that now has become the de facto top one to bolster their defense in free agency. As of last night, they still had lines in the water for safeties Rahim Moore of Denver and Ron Parker of Kansas City. The Giants reached out to the representatives of both players over the weekend.

They also could decide it is worth going after Browns safety Tashaun Gipson, a restricted free agent. If the Giants signed him, they would have to give up a second-round pick in the upcoming draft (he was tendered at that level Monday). And though Antrel Rolle still is unlikely to return, if the market begins to dry up any further and those other players sign elsewhere, the Giants eventually could work their way back to the safety who served as their defensive captain for the last two seasons.

The Giants also are working to re-sign Stevie Brown before he officially becomes a free agent Tuesday. If they cannot, they might be in the position of needing two starting safeties in free agency -- with the consensus best one already off the board and back with the Patriots.

As for Graham, the Giants were more aggressive toward him after losing out on McCourty. They already have put the franchise tag on Jason Pierre-Paul this offseason to stabilize the pass rush, but they clearly were hoping to add more and perhaps reconstruct the kind of defensive end rotation that Steve Spagnuolo utilized in his first tenure as Giants defensive coordinator.

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Other options at the position include Jabaal Sheard of the Browns, Derrick Morgan of the Titans and Adrian Clayborn of the Buccaneers.

McCourty's deal with the Patriots reportedly is a five-year, $47.5-million contract with $28.5 million guaranteed. Graham's deal with the Eagles is reported as a four-year contract worth $26 million with $14 million guaranteed. At about $13 million under the salary cap, the Giants probably could not have afforded those prices (though their interest likely helped drive the numbers up). It's unclear if the Giants actually made concrete proposals to either of them, which would have been against the rules of "legal tampering" anyway.

Notes & quotes: Linebacker Jon Beason agreed to a reduced base salary for 2015, as he suggested he would last week, according to a source. He originally was due to earn $3.6 million for the upcoming season. The new terms were not immediately available, but they will keep Beason with the Giants . . . The NFL Network reported that the Giants have strong interest in running back Shane Vereen of the Patriots. He would join Rashad Jennings and Andre Williams in the backfield if he is signed.