Moore progressed through the Rams academy and up to their U18s squad ahead of the 2021/22 season.
He went on to sign professional terms with the club two years later, and was promoted to their U21s ahead of the current season.
Now his development will be aided by a stint at the Gladiators, who are currently 15th in the Northern Premier League table.
Moore could make his debut on Tuesday night when Martin Carruthers takes his side to play-off-chasing Whitby Town.
The arrival of Moore helps soften the blow of losing top scorer Jonathon Margetts, who joined National League North side King's Lynn Town on Friday.
Margetts had scored 33 goals in 30 games in all competitions since returning to the club last summer following a goal-laden spell which saw him fire Stamford to promotion from Step 4.
Photo credit: Matlock Town
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