Szabó announced earlier this year she would not seek another term after November. She has not ruled out running for Parliament, but has said it is "too early to make a decision".
Earlier this year, Szabó said she was leaving the job "for a range of reasons".
"It's been a big three years and I've done what I came to do. So I think it is a good time to step aside for the next president," she said.
The Party President is usually elected by voting delegates at Labour's annual conference. However this time, no other candidates have been put forward, meaning Day will become President automatically.
Speaking to the Herald earlier this year, Day said her time on Council had been a "whirlwind", particularly becoming deputy mayor under Lester after the incumbent, Paul Eagle left council to become an MP.
She had been a councillor for less than a year.