The Tigers, who went down 3-2 to Wigan in Saturday's third round, have had an indifferent season so far following relegation from the top flight, but still lie within striking distance of the top six.
Barmby said: "We're six points off the play-offs and if we can keep plodding away and we're within touching distance by March then we've got a hell of a chance.
"That'll be down to us now and that'll be our goal, to be in that pack and to strike towards the end of the season.
"We need to bounce back. It's a great derby game to come back to and it's one we'll look forward to.
"It would be silly to start talking about promotion too early. We'll just try and win as many games as we can and see where that takes us."