The London-born marksman, 31, arrived at Edgar Street last summer having taken in 12 different professional clubs during his nine-year career.
He managed to find the net once in eight appearances before November but since then has netted five times in nine, including the opener in Tuesday's 1-1 draw against Crewe.
And Trewick is hoping Constantine can help fire the Bulls up the League Two table.
The Hereford boss told the club's official website: "Strikers thrive on scoring goals and the confidence levels of the players often rise when they get a goal of two so I'm pleased for Leon, he's been working hard.
"He started to get a goal of two before the enforced break and I'm delighted for him. Let's just hope he continues along the same path."
Source: Team Talk
