Saints crashed to their first defeat in six games as the relegation-threatened Saddlers claimed the points after a goal from veteran striker Jon Macken in the 69th minute.
Adkins said: "We have lost a game of football and I'm not going to hide away from that.
"We have to regroup and go again. It was a big blow, but we've dominated the game and I can't fault the application of my players.
"They worked very hard and created a shed-load of opportunities. I've noted 17 down. But it's all about facts, and the fact is we have lost 1-0.
"Walsall battled very hard and rode their luck. We had a hell of a lot of chances, created a lot of opportunities, but there was some inspired defending by the opposition."
Source: PA
Source: PA