Richardson stepped down on Tuesday less than two weeks after leading Stanley to League Two safety, opting to rejoin Paul Cook at Chesterfield.
The 33-year-old previously served as Cook's number two in east Lancashire before the latter made the same switch to the Proact Stadium.
A deal was struck for Richardson to take charge of the Reds but not until Cook made clear his intentions to keep his number two, Heys has revealed.
He told Accrington's official website: "Obviously when Paul Cook left I think there were certainly some conversations regarding him wanting to take Leam with him but I'd like to think we made a really good offer to Leam to stay and take charge.
"That was proven to be the right thing to do because he kept us up and you've got to give a lot of credit for that to Leam. We've reached the end of the season, the conversation's been had again and Leam's decided that his future lies elsewhere."
Nevertheless, Heys admits Richardson's decision to leave caught many at the club off guard.
"We were all together at the end-of-season awards night on Sunday night and I don't think anybody had any idea that this was going to come about," he added.
"It's all happened very, very quickly but it's just one of those things that happens in football. There's not an awful lot we can do about it apart from choose the way we react to it.
"It helps a little bit that we're in the close season. The last two times that we've been without a manager we've had to act fairly quickly to bring somebody in but we can take our time a little bit more.
"It really is an open process. We'll look at everyone who applies for the job and we'll draw up a shortlist, do the interviews and look to appoint the best person for Accrington Stanley."