The former Rangers defender and Scotland international's condition was announced by the club last month.
In a statement on Rangers' official website, Jardine said: "I would like to pass on my thanks to everyone who has contacted my family to pass on their support. I have been truly humbled by the good wishes and it has meant so much to me.
"I also know the supporters arranged a minute's applause at the Elgin game on Sunday and I can't thank them enough for that touching tribute.
"It has been a difficult time for me but I am now out of intensive care and the messages of support I have received have been very uplifting.
"I am truly grateful to everyone who has contacted the club and my family to offer their support."