The 23-year-old shot-stopper has made four appearances for the Cumbrians since joining the club in November.
Amos, who will remain at Brunton Park until January 1, 2014, wants to help the team make further progress over the Christmas period.
He told the club's official website: "I've thoroughly enjoyed the last month because it's been a chance for me to get onto the pitch and get games.
"I'm delighted that it's been extended because this has been a great move for me.
"The fans have been brilliant with me and we've picked up some decent results along the way. The games come thick and fast over the Christmas period and hopefully I'll do enough to be involved and to help the team even more."