The south-coast side has been on a turbulent ride since relegation from the Premier League in 2005 and three years ago was on the brink of bankruptcy.
However, the club survived and moved on, with back-to-back promotions putting it back in the top tier.
It kicks off its campaign at champion Manchester City on Monday morning (AEST) and Adkins will not be taking life at the top for granted.
"We want to be very ambitious and have said that from the start," he said.
"The important thing, having just got back-to-back promotions and the football club having been out of the
"(In a bid to stay there) we worked very hard through the course of the summer to identify players.
"I think when you look at the Euros and Olympics then you probably find that a lot of transfers throughout the course of the summer have been very slow.
"A lot of things have been waiting and growing behind the scenes and normally it is in the last couple of days where the majority of the transfers are done.
"I would like to think Southampton would be one of the clubs that may make one, two or three more additions into the group that we've got."
As well as adding wide men, the Saints boss has made no secret of his desire to shore up the central defence, but has been thwarted so far.
"I'd like to get one in to start off but if we can get two even better," Adkins said.
"I wouldn't say I have been frustrated, this is what football is.
"Our endeavour is to keep improving and the reality is you target players. Maybe there is only a certain calibre of players you are going to target and other teams are going to target as well.
"Sometimes you can get the players, sometimes you can't. I know Southampton works very hard in the different departments to make things happen.
"Sometimes they don't and then you just have to move on to the next one."
English Premier League
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