Adkins suffered defeat against his former club on Saturday as goals from Jay Rodriguez and Adam Lallana left the Royals seven points adrift of safety, with just six games of the season remaining.
He admitted after the final whistle that Reading had needed to win, but says the club have no choice but to keep battling for the remainder of the campaign.
"We'll pick up the positives we can take from it and we'll work very hard on all the aspects of our game to be ready for the next match," Adkins told Sky Sports.
"We have to be positive. I'll pick the players up, the players will pick themselves up and we'll work very hard.
"This is the reality of it and you have to embrace that challenge. It is a situation we don't want to be in but we are in, so we can either start hiding away from things or face up to the reality of where we are and look forward to what we have got to go and do."
Reading started brightly at the Madejski Stadium but spurned a glorious opportunity to open the scoring when Adam Le Fondre blazed over.
Adkins refused to criticise his striker, though, saying: "Alfie is the leading scorer, he is a great finisher and we have got him on the field of play to score goals when the opportunity comes.
"Unfortunately it didn't happen today but I am sure it will happen in the future."
Southampton's opener came against the run of play when Rodriguez beat Adam Federici to the ball in the 34th minute, but Adkins was not sure the goal should have stood.
He explained: "The challenge on the goalkeeper, I think maybe somewhere else on the field could have very easily been given as a free-kick. His foot is very high.
"It was very pivotal because it put Southampton 1-0 up and I thought we had been doing well in the first half. Unfortunately, Southampton got a better grip of the game in the second half."