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Earlier this month, the Waitaki District Council-controlled company announced that Mr Swann planned to stand down at the end of June from the role he had held since January 2017.
Waitaki Mayor Gary Kircher said a new chairman would be voted in by board members by the end of this month and start in the role on July 1.
Whether that person would be installed on an interim or long-term basis was a matter for the board to discuss, he said.
"They've got to vote and have that discussion ... and whether they need an interim chair and revisit that in six months' time, or do they go forward."
Expressions of interest for positions on the board will be called for later this year and ratified by the council in late November or early December, after October's local government elections.
"It will allow the dynamics of the board to settle down and for us to be able to clearly identify the skills needed without rushing into it," Mr Kircher said.
Those appointments would be effective from January 1.
On the day of the announcement, Mr Swann told the Otago Daily Times the time was right to step aside after a two and a-half year period of ongoing change at the hospital, which included the completion of a joint review of the district's health services carried out with the Southern District Health Board in late 2016, and more recently the controversial proposal for change review.