The source predicted Smith would miss at least "a few more" games, though another person said the receiver might be able to return against Washington the first Sunday in December.
The Giants initially categorized Smith's injury as week to week, but the truth is it is closer to a month-long injury and could turn out to be longer depending on Smith's pain threshold and doctor's evaluations.
The Giants also lost Smith's replacement, Ramses Barden, to a season-ending ankle injury that left him on crutches and in a cast Monday.
But Tuesday the team signed Derek Hagan to a one-year contract for the league minimum, according to a league source. Hagan played for the team last season, primarily on special teams, but was cut in training camp.
Boss, however, suffered a lower back sprain in the game against the Cowboys. He hopes to play Sunday night against Philadelphia.
The Giants recently waived Sinorice Moss from the injured reserve and the team says Moss is not eligible to return. Rookie receiver Victor Cruz was placed on injured reserve earlier this month with a hamstring injury. The Giants, who have wide receiver Samuel Giguere on the practice squad.
Adam Schefter is an ESPN NFL Insider. Information from ESPNNewYork.com's Ohm Youngmisuk and The Associated Press contributed to this report.