The Serbia international signed on a free transfer in the summer but has failed to impress, scoring twice in 17 appearances.
Jovanovic has started just five Premier League matches, the last coming in the 2-1 defeat at Blackpool a fortnight ago - his only appearance in Dalglish's squad since the Scot replaced Roy Hodgson earlier this month.
The 29-year-old was linked with a swift exit in the current transfer window but he is hoping to get a second chance under the new regime.
"I am sure the new manager coming in has given everyone a lift, those who are playing and those who aren't," he told LFC Weekly.
"I think the second half of the season can be much better.
"When I first came here I expected things to be better. It hasn't been good enough so far but I am strong and I keep working hard.
"I don't ever think to myself: 'I deserve to play or I don't deserve to play.'
"I just keep going and working hard. Sometimes a chance can come when you least expect it. That is life so you have to be ready."
