What is the connection between women's influence in the public sphere and the influence of evangelical religion on society?
To a degree, the answer to this varies with period. Although many twenty-first century evangelical groups seem opposed to full equality for women, that was not always the case.
Various eighteenth- and nineteenth-century Protestant evangelical groups allowed far greater equality for women and public activity for women than more tradition Roman Catholic and high church groups. The first reason for this was the notion of the "priesthood of all believers" and the "right and duty of individual judgment". Rather than Scripture and religion being mediated by a male priesthood, in evangelical circles individuals related directly to God. Women in these communities could act as preachers.
Also, the universal Christian duty of evangelism obliged women to speak out on moral issues publicly as well as privately. Both the abolitionist and temperance movements had many female leaders who spoke publicly and published tracts.
An interesting fact is that the Salvation Army was one of the earliest Christian movements in which all ranks were equally open to males and females, and female leadership was common.