The ring was put under an export bar by culture minister Ed Vaizey following the July auction, giving UK institutions and collectors until September 30 to match the £152,450 price paid by the US singer.
Vaizey states that the ring should stay in Austen's country of birth due to its historical importance.
The museum now has until December to raise the remainder of the sum, with fundraiser Louise West explaining to the BBC: "We knew it was going to go for much more than we could raise in that short space of time."
The museum has invited Kelly Clarkson to visit the house, which is where Austen wrote all of her six completed novels.
The auction record for any Jane Austen item is held by the unfinished draft for her final novel, The Watsons, which sold for $1.6m at Sotheby's in 2011.
Paul Fraser Collectibles has a fantastic stock of rare literary collectibles for sale.