Hartlepool have announced they will not sign convicted rapist Ched Evans after boss Ronnie Moore expressed an interest.
Arsenal manager Arsene Wenger insists he will only spend money in January on the right player.
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's defeat to St Mirren, which saw Hearts slip to second bottom of the Clydesdale Bank Premier League.
The club's financial problems have been well documented in recent months but Southern believes results on the park should be better despite cut-backs.
He said: "Hearts, I believe, have the second-highest playing salary bill and we should be performing much better than we are.
"For Hearts to be sitting in 11th place, the results don't lie.
"What we have at our disposal are some very talented young players and, indeed, John played his role in bringing those young players through to the first team.
"However, we shouldn't forget there are also senior professionals at this club. They also have a role and responsibility to the club.
"The secret will be to merge and blend those senior players with the young players to establish a successful team and a team with a winning mentality."
McGlynn told Hearts News: "I was very grateful to be given the chance to become the Hearts manager as it was my dream job.
"It was an honour and privilege to serve the club in this capacity and I thoroughly enjoyed working with the players, who have been top professionals.
"It is unfortunate that things didn't turn out the way I wanted them to and I am gutted about that.
"I am a worker and want to get back into football as quickly as possible.
"I don't think I could have done any more, but the league table doesn't lie and I know it is a results-driven business.
"We had a lot of difficult circumstances to deal with and they have taken their toll in the end, but no-one will be more delighted than me if the team can go on and win the League Cup."