YouGov survey suggests backing for the left-winger among Labour members is only increasing.
Smith said he entered the race as a representative of Labour's new generation.
Angela Eagle has faced attack since launching leadership challenge to unseat Labour leader Jeremy Corbyn.
Embattled far-left leader Jeremy Corbyn will contest the challenge while the Conservatives are poised to choose between Theresa May, Andrea Leadsom.
Blair's shadow may hang over the party but challengers to Corbyn should still look to his example to win.
Leader Jeremy Corbyn is facing an exodus from his top team.
"He's the life of the party, but he's not who you want driving you home at the end of the evening."