Cantor, who lost his re-election bid in Virginia's 7th District to his primary opponent, pressed on Gov. Terry McAuliffe for a November 4 election in his district to coincide with the general election.
By taking office in November, rather than wait until January, "he will also have seniority, and that will help the interests of my constituents (because) he can be there in that consequential lame-duck session," Cantor said.
The seven-term congressman, who was largely expected to become speaker of the House when John Boehner, R-Ohio, retires from the role, officially stepped down from House leadership at midnight Thursday. He will resign his seat effective August 18.
On Thursday, Cantor endorsed Dave Brat, the economics professor whose surprise victory in the June 10 primary upended Cantor's political career.
"I hope he will win," he said.