Saints start their Europa League campaign on July 18, more than two weeks before the start of the Scottish league season. Having seen the likes of Murray Davidson move on after the end of the season, Wright is looking for reinforcements.
But he will make sure any new player fits into a team that has qualified for Europe two years in succession.
Wright, who has stepped up to replace Steve Lomas, said: "We're not panicking. We won't rush into bringing players in.
"We'll do our homework on them and make sure they are right for St Johnstone. We pride ourselves in team work and being a unit that works really hard for each other.
"We have 16 signed players at the minute which is three or four more than we had at this time last year. We won't rush players in just for Europe, we will look at the bigger picture for the league campaign that starts in August."
Another potential problem is the continuing uncertainty over the future of Richard Brittain, who signed a pre-contract agreement with Saints in January but then declared his wish to stay at Ross County.
County reportedly expect Brittain to begin pre-season with them but Wright still wants the player.
"I feel quite rightly that he's our player and he should turn up at pre-season on the 27th," Wright said. "I am looking forward to seeing Richard here. We watched Richard a lot in the First Division and also tried to sign him then but the deal fell through.
"We certainly hope he is going to come but it might be out of my control. The chairman is dealing with it and I trust him to do the best for St Johnstone."