The Bosnian-born shot-stopper, 29, has made a total of nine first-team appearances for the Premier League club since arriving in the summer of 2012.
The last of those came in a 2-2 draw at Tottenham in the Capital One Cup at the end of October, while he twice faced the O's last season in the FA Cup third-round meeting between the two sides which saw Hull progress following a replay.
Jakupovic's arrival comes after Orient's first-choice goalkeeper Jamie Jones was recently ruled out for up to three months with a hairline fracture in his back.
Jakupovic, who has one senior international cap for Switzerland and was selected as part of their squad for Euro 2008, has previously spent time playing in Switzerland, Russia and Greece.
He has found himself behind Allan McGregor and Steve Harper in the pecking order at the KC Stadium this season.