Jacksonville had cut Mincey last week over repeated issues of tardiness and violation of team rules.
The 6-4, 265-pounder was drafted by the Patriots in 2006, and played under Jack Del Rio for 4.5 seasons with the Jaguars.
In 243 snaps over eight games this season, PFF has credited Mincey with two sacks, a QB hit, 16 hurries, 12 stops, and a grade of plus-3.4.
Officially, he's played in 62 career games (33 starts), with 20 sacks (including eight in 2011), one pick, 11 passes defensed, six forced fumbles, and 119 tackles.
Correction: Article originally stated that Jaguars had drafted Mincey; rather, it was the Patriots