I don't know what time period you're referring to, but most freed slaves in the American South did not own slaves of their own. There are a few famous examples of former slaves who ended up making their own plantations, complete with slaves, but they were definitely the exception rather than the rule. Most free blacks remained fairly low class for their lives, maybe becoming small farmers if they were lucky. Many free blacks became day laborers or sailors and drifted into cities where it was easier for them to find work. Of course, if you're talking about slaves in the Roman Empire or something, I don't really know since it's not my area.

Sources: Max Grivno, Gleanings of Freedom

W. Jeffrey Bolster, Black Jacks

Ira Berlin, Slaves without Masters