Michael Beale's side, who trail city rivals Celtic by nine points in the cinch Premiership, have a Viaplay Cup semi-final against Aberdeen to look forward to on Sunday before kicking off their Scottish Cup campaign away to St Johnstone the following weekend.
Jack, who scored in last season's Scottish Cup final triumph over Hearts, knows the importance of trying to put silverware in the trophy cabinet in the second half of what has been a challenging campaign.
"That's the aim for me and the team and the staff," said the midfielder. "Since the manager has come in, he's said we want to go win both cups.
"Obviously with me being here for a number of years, I know the demands the fans put on us to go and win every trophy.
"The boys will be ready for next week and it's a chance for us to get back to Hampden and another cup final.
"We're expected to go and try to win every competition we're in. The supporters demand it, the manager and the staff demand that from us.
"But for us as players, we want to go win as many trophies as we can. It is something to look back on at the end of your career.
"Obviously last year we won the Scottish Cup and we want to win both cups again this year too.
"We'll work as hard as we can this week and we look forward to the game."
Jack believes the return of several players from injury will help Rangers in the second half of the season, with Kemar Roofe back in the squad for Sunday's 2-0 win at Dundee United and both Ianis Hagi and John Souttar close to making their comebacks.
"When we've got a full squad fit then there's no doubt we've got unbelievable talent in this squad," he said. "We've just not had that often enough this season.
"Now that we're heading into the back end of the season, we want to get everyone fit and get everyone firing in training first and foremost.
"Once everyone is up to speed that will take us where we want to go."
Jack is out of contract in the summer but insists there has been no notable movement with regard to a potential extension.
"No, nothing has changed since the last time I spoke to the media," he said. "Everything is still the same. There's been nothing.
"Look, I've said it before, I just want to play as many games as I can and work as hard as I can. When the time comes, we'll sit down and we'll speak.
"But it's not the time yet. I just want to focus on doing well for the club, trying to play every week and staying fit."