Change in social status was a factor in the growth of early Christianity. However, status itself is never simple and is a composite of several economic, cultural, and political factors.
For instance, trademen could rank quite high in Roman culture economically, but would not rank as high culturally, not being tied in to Greco-Roman arts and tastes. And someone as socially low in status as a slave could become a philosopher or a successful business owner, but would always carry the social stigma of slavery.
What is true of Christianity in its early days is that is set about equalizing the social status of key groups of individuals. That is to say that a slave or former slave carried no social stigma in Christian belief- one can see where this would be compelling for a freed slave trying to make his or her way in a Roman world and never quite getting there.
Similarly, though women could be wealthy and manage households under Roman custom it is only in Christianity that their sex carried no social disregard- and it precisely the women who brought Christianity into the Roman upper classes.
And though the army would only later become an engine for the spread of Christianity, it was key as another place where social status could be leveled out- Roman soldiers could earn high respect but never be wealthy; Christian communal practice of sharing wealth and seeing great wealth as an impediment to spirit might have done well to help their conversion along.
So it is not surprising that Christianity spread in precisely the places where these status inconsistencies proliferated: in the Roman household, in the Greco-Roman club, and in the tradehouses and places of work in Roman cities. Rather we should only note that it spread as swiftly and freely as an illness or contagion of those existing Roman institutions.
These were the places of relationship in Roman society, and Paul went there if he failed to succeed in the initial Jewish synagogues. Indeed, the most significant thing about Roman cities is their lack of private space. Extremely dense even by modern standards, every urban place was public or semi-public. Word- of anything- thus spread rapidly. Initial contacts spread the word- to everyone, including those who most wanted to hear.
Another key factor in spreading the contagion of early Christianity would have been an ethnic community internal to each Roman city- especially the Jewish sections. A new convert would thus be even more immediately connected to members of club, broader family and fellow practitioners of his trade.