New Zealand Cricket confirmed it will advertise the role after Stead chose to end a tenure in charge of the ODI and Twenty20 formats which began when he replaced Mike Hesson in 2018.
NZC says it hasn't yet determined if it will seek a coach to again oversee all three formats or if it will split the role, as is common in other countries.
Stead, 53, said he will decide "in the next few weeks" whether to stay with the test side but conceded he wasn't motivated to coach across all formats.
His contract was up for renewal in June.
"I'm looking forward to getting away from touring life for a while and having a think about my future," Stead said in an NZC statement.
"The past six to seven months has been particularly busy with relatively non-stop cricket action since September.
"I now want to evaluate my options but still feel I have coaching left in me, albeit not as head coach across all formats.
"The next month will give me the opportunity to discuss the situation more with my wife, family and others.
"I'll be in a better position to know whether I want to reapply for the test coaching position after this time of reflection."
Stead has largely continued a period of success for the Black Caps that had begun under Hesson, resulting in his contract being renewed: in 2020 and 2023.
The former Canterbury opening batsman, who played five tests, has overseen some exceptional performances on the test stage as a coach.
He helped to steer the Black Caps to the inaugural world test championship title in 2021 and to a historic 3-0 test series sweep of India on the sub-continent last year.
NZC chief high performance officer Bryan Stronach said Stead deserved the opportunity to take some time out over his decision.
"Gary's results have been very impressive over a long period and we're very comfortable giving him some time to collect his thoughts and mull things over," Stronach said.
"At the moment we haven't any strong preference for either a split-coaching role or a sole appointment who covers all three formats, and we're unlikely to be clearer on that until we see who's putting their name forward."
