McGlynn parted company with the Edinburgh outfit with immediate effect on Thursday after less than a year in the Tynecastle hotseat.
First-team coach Gary Locke and player development manager Darren Murray will take interim charge of team affairs ahead of the visit of Motherwell at the weekend.
Hearts chiefs have now launched the search for their next manager and Southern says they are determined to ensure they make the right appointment.
He told HeartsTV: "We don't want to categorise anything in terms of who that manager might be, where they might be, whether they'll be with a club or not with a club or what their nationality might be.
"We also don't want to box ourselves in in terms of timescale. We are very aware that there are some very important games coming up.
"We're assured by Gary Locke and Darren Murray that we're in safe hands for the immediate future with both those chaps.
"They took the first training session this morning and will carry on taking training sessions through to the Motherwell game and maybe even beyond. We will not rush the job of appointing a new manager - it's absolutely crucial we get it right."
He added: "Potential candidates should be fully aware of the scale of the job they're taking on and the size and importance of Heart of Midlothian Football Club to so many people."
McGlynn's departure followed Wednesday night's defeat to St Mirren, which saw Hearts slip to second bottom of the Clydesdale Bank Premier League.