According to ESPN's Adam Schefter, Wilcox "flipped" at the last minute when he had already made a trip to the 49ers offices to sign a one-year deal.
"He got stage fright," a source told Schefter.
The 27-year-old appeared in 12 games last season and finished with 12 tackles, one interception and one pass defended.
Wilcox originally signed a two-year deal with the Tampa Bay Buccaneers last offseason, but he was traded to the Steelers for a 2018 sixth-round pick. The Steelers also received a 2019 seventh-round pick.
Although Wilcox never became a major contributor in Pittsburgh, he was a key member of Dallas' secondary for several seasons. In four years with the Cowboys, Wilcox made 38 starts and registered five interceptions.
His best season came in 2014 when the 2013 third-round pick out of Georgia Southern finished with 81 tackles, three picks and five passes defended.
With Jamal Adams and Marcus Maye penciled in as the Jets' starting safeties, Wilcox will likely be a backup for Gang Green in 2018.