The special election for the seat will take place during November's General Election.
The bill won Senate passage on a 30-5 vote after an hourlong debate.
U.S. Rep. John Yarmuth spent eight terms representing the Louisville-area 3rd District, officially retiring from the post on Jan. 2.
It's not known how long the speaker vote process will take.
Yarmuth, a Louisville Democrat who represents Kentucky's Third Congressional District, announced in October 2021 that he would be retiring.
Jefferson County Judge Annie O'Connell ruled Wednesday that Susan Tyler Witten, who was on the ballot for the 31st House District, would be disqualified.
Kentucky lawmakers wrapped up redistricting work in a rare Saturday session, passing bills to redraw congressional and legislative maps that now will be scrutinized by Gov. Andy Beshear.
The governor's agenda is organized into five pillars.
The 77-year-old Grooms said it's time to pass the baton to younger leaders.
The bill would replace a process that allowed the governor to appoint someone to fill a Senate seat until the next regular election of the U.S. House of Representatives — every two years.