It was very common for Roman freepeople to become slaveholders themselves. What you need to keep in mind is that while manumission was common in Rome, it would not have been granted to the vast majority of the enslaved population. Usually only those closest to the family or who were skilled in some way would be manumitted. Those who labored in the worst conditions, like in mines or on large farms, would not have been freed (this is not to say that other enslaved people did not experience violence or bad conditions). If the freed person was continuing in some sort of business for their former enslaver, they would need enslaved labor to help with that. Many freed people would have had slaves in their households simply as a mark of status and success.

The most famous example of a freed person in Roman literature is probably Trimalchio in Petronius's Satryicon. He is an exaggerated example of a freed person who attained wealth (and consequently a large number of slaves). This did not mean that freed people perhaps did not see themselves as "better" than their former enslavers (not that they were, but as a self view). Trimalchio describes how he plans to free all his slaves in his will, showing some sympathy for the condition of slavery: "Ah, my friends, a slave is a man and drank his mother's milk like ourselves, even if cruel fate has trodden him down. Yes, and if I live they shall soon taste the water of freedom. In fact I am setting them all free in my will. ... I am making all this known so that my slaves may love me now as it I were dead" (Satyricon 71, trans. Michael Hesetine). The joke here is that Trimalchio wants funeral-level praise while he is alive, but you can see that he talks about slavery as an unlucky turn of fate. He has at least some sympathy for slaves, but he still works within the Roman slave system.

Aside from exaggerated satiric examples, we have many, many inscriptions from freed people that mentions their slaves or own freed people. For example, in Ostia, a freedwoman named Scribonia Attice made a tomb dedicated to her immediate family, but also to her freed people and their descendants (Tomb 100, see ostia-antica site for the inscription and images). She would have had her own slaves as part of her midwifery business. Scribonia Attice is just one example of a common practice. Freed people likely saw slaveholding as a status symbol and/or a necessary aspect of their work. Since they were a "good" slave and got manumitted, they may have been less likely to view the system as a problem, but as one that could be worked within.

A good overview of Roman freedman is:
Treggiari, Susan. 1969. Roman freedmen during the late Republic. Oxford: Clarendon Press.