Tempest, 25, is a former Northern Ireland Under 21 international who made 16 appearances for Notts County between 2012 and 2015 before moving to Lincoln City after loan spells with Ilkeston and Boston United.
He then played for Nuneaton and at Gainsborough Trinity on loan, eventually signing for Basford during the summer.
Tempest will be added to the Matlock squad for Saturday's trip to Witton Albion.
Harrad’s move in the other direction was confirmed earlier, after he was released by Matlock following the end of his loan spell at Grantham Town.
Matlock have also released midfielder Harry Vince, having been out with an ankle injury picked up in the win over Nantwich Town at the start of November.