The Tigers have spent the majority of the season in and around the halfway point in the league and despite lying 13th, they remain just three points clear of the relegation zone.
Manager Steve Bruce has consistently said survival is the primary goal of the season, but victories over Middlesbrough and Southend have taken his side within sight of the an FA Cup quarter-final.
To achieve that, they need to see of ambitious second-tier opposition in the form of Brighton on Monday night.
Bruce junior accepts the importance of retaining top-flight status but believes the club can successfully fight on two fronts.
"Obviously whenever you play you want to progress to the next stage, especially when you're in sight of a quarter-final," he said.
"The league is the most important thing but to have a chance of going to Wembley would be massive, the kind of thing you work hard for all your life.
"If we can stay up and do well in the cup that is the perfect situation. Teams take confidence out of winning football matches, regardless of if they are in the league or the cup.
"A win on Monday brings something you can take into the next league game, because it's about playing well and developing that winning habit."
Bruce also has a personal reason to defeat Brighton, with his own FA Cup CV in need of some improvement.
Father Steve was a three-time winner with Manchester United, but Alex has yet to come close in his own experiences with Birmingham, Oldham, Ipswich and Leeds.
"I've got a crap record in the FA Cup - fourth round, fifth round, never as far as a quarter-final," he said.
"This run with Hull is as good as it's been.
"If we can carry on there could be a fantastic occasion for everyone concerned."
The Tigers have made major changes for every cup match this season, in a bid to rest regulars and satisfy those short of game time.
That policy is expected to be eased somewhat against the Seagulls, with captain Curtis Davies expected to make his first appearance in the competition this season and the likes of Maynor Figueroa, Jake Livermore, Allan McGregor and Ahmed Elmohamady also due to start.
Definitely absent are January signings Nikica Jelavic and Shane Long, cup-tied with Everton and West Brom respectively.
In their stead Yannick Sagbo and Matty Fryatt will hope for a chance, while Sone Aluko should continue his comeback from four months of Achilles trouble at some stage in the game.