Best known for successful spells at Mansfield Town and Torquay United, Briscoe is equally as comfortable on the wing or in central attacking positions.
The 27-year-old, who began his career with Port Vale, won promotion from the fifth tier with Mansfield in 2012/13 and was a runner up in the FA Trophy with the Stags in 2011 - the season he was also voted player of the season at Field Mill.
Briscoe joined Torquay from Mansfield in June 2014 and scored 9 goals in sixty-six league and cup appearances at Plainmoor.
He had his contract cancelled by mutual consent in January 2016, and joined Wrexham until the end of the season.
Briscoe has featured as a triallist during pre-season, and player-manager Andy Morrell said: "Louis has plenty of experience. He is hungry and wants to do well. He ticks all of the boxes".