The Toffees midfielder returned from a knee injury against Fulham and is keen to fight off competition for his place in the team
Gibson returned to action in Everton's 2-1 Capital One Cup defeat against Fulham on Tuesday, completing 70 minutes before being substituted.
The 25-year-old had been out of action since August 11 when he sustained a knee injury during a pre-season friendly against Real Betis and, having returned, he is hoping to remain in the team.
"I've come back fit and strong so, fingers crossed, I'll stay injury free now," he told the club's official website.
Everton boss Roberto Martinez hailed Gibson as a "vital player" for the Toffees following the loss against Fulham and the former Manchester United midfielder is pleased with the recognition, but acknowledged that he is fighting for his place in the team.
Martinez bolstered the Toffees' midfield options in the summer with the addition of Gibson's compatriot James McCarthy as well as England international Gareth Barry, and the Derry native has welcomed the increased competition.
"It's nice to hear the manager say things like that," said the Ireland international. "He's brought new players in and obviously there is a bit of competition in midfield now.
"That's good for the squad. It makes everyone try harder in training and try harder in games. Everyone is fighting for a place."