Adkins had also secured consecutive promotions with the St Mary's outfit and McMenemy, who led Southampton to their highest ever top-flight finish and an FA Cup victory during his tenure, has labelled the decision as "a terrific risk".
"It's not as if they have brought [Jose] Mourinho or [Pep] Guardiola in," McMenemy told BBC Sport."They have brought in a chap - with due respect he might be a nice man and a very good coach, but who nobody has heard of.
"It is a terrific risk. We don't know anything about the man coming in. We don't know what reaction the players will have.
"Normally, when a manager gets the sack it is because things have gone wrong but not on this occasion. Timing is everything in sport and, at the minute, it doesn't look like the time of this decision was right. But time will tell."
You can follow Southampton vs. Everton - Pochettino's first game in charge of the Saints - live here.