The Robins parted company with Maurice Malpas last week after less than 12 months in the County Ground hot-seat.
The Scot was the club's sixth manager in just over three years and Fitton has vowed to bring stability to the League One side with his next appointment.
"I didn't want to be a chairman agreeing that a manager should go after less than a year," he said.
"I genuinely hope the next appointment is for a long time. It is really important that the person who comes here wants to be here and wants to do something for Swindon in the long term.
"We have to accept that some people will use this as a stepping stone. I can accept that, but if we are going that way then they have to produce results too."