The former Championship club on Saturday suffered a 1-0 defeat at Anfield, where Daniel Agger's header late in the first half proved to be the difference.
But Adkins was pleased with his team, who surrendered a four-game unbeaten run in the Premier League which had followed a previous record of eight defeats in 10 matches.
The Southampton boss thought there were positive signs in his relegation-threatened players refusing to sit back despite being dominated by Liverpool for long periods.
"That is probably as good movement as we have played against this season," said boyhood Liverpool fan Adkins.
"So from our point of view, we have come a long way, because we want to get points but we have managed to stay in the game."
He added: "The players are improving all the time. There is a lot of belief about ourselves and we look forward to the next games."
Agger's header came in the 43rd minute and Adkins was disappointed with the timing of the goal.
"We pressed them towards the end of the game, which was good," he said. "There were a few 'oohs' and 'aahs' in that respect.
"A goal just on half-time, from our point of view, obviously, we would have liked to have gone in at half-time at 0-0."