The 20-year-old has dropped further down the pecking order at The Jakemans Stadium following the arrival of Liam Wakefield and moves to The Northolme in pursuit of regular first-team football.
Hare's temporary departure comes at the end of a week which has also seen striker Tyrell Waite join Evo-Stik League Premier side Matlock Town on loan for one month.
The 23-year-old made his debut in Tuesday evening's 2-1 defeat at Barwell.
Defender Brad McGowan completed a permanent move to The Jakemans Stadium from Alfreton Town earlier in the week following a successful loan spell with the Pilgrims.
The 21-year-old scored twice in his initial month-long spell at the club and is now with the Pilgrims on a contract until the end of the season.