Jones has started his tenure on the south coast with two losses from two matches after taking over from Ralph Hasenhuttl following his dismissal in November.
Following their late defeat to Fulham on New Year's Eve, the Saints entertain fellow strugglers Nottingham Forest in a pivotal fixture at St Mary's Stadium off the back of five consecutive league defeats.
"I know it can turn sour quickly and I don't want that to happen," said former Luton boss Jones.
"I know what we're good at, I know what improvements we've made, or are trying to make, and I know the levels we're producing in training and in games. Now, we have to make sure we continue that.
"It's not going to happen overnight, but it needs to happen quicker than it is, because time will run out eventually.
"But you can't change the absolute direction of a football club in a few weeks. With the greatest respect, I want to manage expectations here."
Despite their desperate run of form, Southampton could climb out of the bottom three if they earn their first home win since defeating Chelsea in August on Wednesday.
Forest are also in the drop zone, two points clear of the Saints, and have not tasted victory on the road this season.
However, Southampton are the league's worst-performing home side, taking just six points from 24 and Jones has warned the St Mary's faithful must stay on side to help his stuttering squad.
"There's no doubt the players know the importance of the game, and of the situation," added the Welshman.
"We have to make sure that we're the best version of ourselves to give the fans something to get behind, and they do really need to get behind us because we haven't won enough games this year.
"Moments happen in football that can change confidence, change momentum, change everything. When you start well and get the first goal that can instil confidence.
"And the crowd have a big part to play in that. If there's negativity from the crowd, or background noise, it can have an adverse effect on players.
"We've got a lot of younger players in the squad that, at times, need help, and if they get that help they could be wonderful players.
"But there's a process to get into that. And, right now, we have just arrived in the middle of that process. So, everyone's in this together."