Edgar, the current coach of the Wellington Firebirds, told the Herald on Sunday two weeks ago that he would not be applying for Hesson's position - adding to a growing list of high profile New Zealanders ruling themselves out of the running.
Northern Districts and former national coach John Bracewell, T20 specialists Stephen Fleming and Daniel Vettori, and incumbent Black Caps assistant Craig McMillan are also not expected to apply.
Calls placed around the cricketing traps gravitate back to Canterbury coach Gary Stead, Central Districts coach Heinrich Malan and Scotland and former Northern Districts coach Grant Bradburn being qualified contenders.
Stead has guided Canterbury to three Plunket Shield titles in the past five seasons and the 2016-17 Ford Trophy.
The 46-year-old has also worked as an assistant during Hesson's tenure and coached the White Ferns, meaning he is well-acquainted with NZC structures.
Malan helped CD to last season's Plunket Shield and back-to-back Ford Trophies in 2015 and 2016. The 37-year-old previously told the Hawke's Bay Today he would wait until a job description was released before making a decision.
Bradburn has previously indicated his intention to apply for what has been a long-coveted role.
The 52-year-old's contract with Scotland finishes this year. His 2008-13 Northern Districts' tenure returned two Plunket Shields and two List A titles.
He has coached the New Zealand A and under-19 sides and is now working as an assistant to John Wright at Derbyshire in England's domestic T20 competition.