Belgian Evenepoel pulled out of the race on Sunday, shortly after regaining the overall lead with a tight victory in the time trial after Colombian Rigoberto Uran (36) and Italian Filippo Ganna (26) were also ruled out with the virus.
"We relaxed our attention too soon," Vegni told Italian sports daily La Gazetta dello Sport on the Giro's first rest day.
"We cannot let our guard down. As early as this week we will put back some restrictions that had been abolished, such as the obligation to wear masks in the areas where you come into contact with the riders.
"Should we have done this earlier? Probably yes."
Teams are not required to test their riders for COVID-19 and even in the eventuality of a positive case, they are not obliged to pull the rider out of the race.
Britain's Geraint Thomas (37) will start Tuesday's 10th stage as the new race leader.