The Edinburgh club went six matches without losing before going down 3-0 against league leaders Caley Thistle last weekend.
Hibs face a tough run of fixtures which begins on Monday with a trip to Firhill to face a Partick Thistle side who have only lost twice in the Scottish Premiership.
Fenlon's men then take on the current top four in the league - Celtic, Aberdeen, Motherwell and Inverness - with a League Cup quarter-final against Hearts in the middle.
"We want to go as far as we can," Fenlon said. "We went on a six-match unbeaten run before we lost at Inverness and we want to make sure we go on another unbeaten run.
"We have some real tough games coming up but we need to be competitive and go and win games against teams ahead of us in the league."
Thistle have the incentive of going above Hibs if they win on Monday night and Fenlon has warned his players that they face a difficult encounter.
"They are a good side," the Irishman said.
"I have seen quite a bit of them but I had the chance to see them in person against Dundee United in the League Cup.
"They were excellent on the back of a poor start. That might sound strange when they lost 4-1, but the scoreline didn't reflect the game. They passed the ball around well.
"They have a really good shape and they know the job they want to do and how they go about it."