At this point of the offseason, it's pretty impossible to make any big-name signings. With the trade deadline gone and free agency not opening until March, only players who were not under contract at the end of the regular season can be signed to futures contracts, so you're not likely to find many household names.
One additional possibility is anyone who was on the roster of a playoff team but then got released after the end of the regular season. The waiver period for such players opens on the day after the Super Bowl and the Jets would be third in the waiver priority based on their regular season record.
That scenario only applies to one player this season, but it's a player the Jets have shown interest in before. Back in March the Jets were interested in signing Ron Parker after his release from the Chiefs, having also been in the mix for his services back in 2015. In the end, he got signed by Atlanta in June, was released by them in final cuts and ended up back with the Chiefs, for whom he started 14 games and played over 90 percent of the defensive snaps.
Towards the end of the season, Parker was a healthy scratch and ended up being released so that the Chiefs could activate Laurent Duvernay-Tardif from injured reserve. He graded out reasonably well during the season though and brings good experience and versatility, with the ability to play in the slot or as a deep safety.
At 31, Parker wouldn't be a long-term addition but could be a valuable depth addition or at least someone to throw into the competition for roster spots.
He's actually out of contract at the end of the season anyway, so claiming him off waivers wouldn't guarantee the Jets anything. However, it wouldn't cost anything or cause them to make any kind of commitment and what it would do is give them the inside track on being able to sign him to a deal before or shortly after free agency gets underway.
The Jets did exactly the same thing with Mike Pennel two years ago, claiming him on the day after the Super Bowl even though his contract was up and then signing him to a one-year deal in March, so it's an approach they have used before. The other reason a team might do this would be to collect on a compensatory pick once the player signs for someone else, but the Jets figure to be acquiring more free agents than they let go, so that may be a moot point.
Parker is a player who has been released 11 times in his career, but he's started 76 games in the last five seasons and is not really the kind of addition that would excite the fanbase, but he could be viewed as a potentially useful role player for a team that has suffered over the past few seasons when they've had to give reps to inexperienced players in the slot or at safety.
The Jets' previous interest in this player might have come from the (now replaced) coaching staff rather than the front office, but with no real downside and nothing to lose, this could still be a move the Jets consider, so be on the lookout for news on this early next week.