Shadow Welsh Secretary Mr Smith entered the fray this week and will stand against Mr Corbyn and Angela Eagle.
The Pontypridd MP, who cancelled his campaign launch in the wake of the attacks in Nice, has been given a boost after winning the support of Cardiff Central MP Jo Stevens.
Ms Stevens previously had stood by Mr Corbyn, but has now told Labour members via e-mail of a change in her allegiance. She wrote: “During the past two weeks it has become painfully obvious that we have been unable to fulfil the very basic day to day operation as the official opposition in Parliament.
“We cannot present ourselves as a government in waiting without a leadership team that commands the respect and support of not only members but Labour voters.”
She added: “I want to let you know I will be supporting Owen Smith.”