Need advice on top soil for laying sod?

We just got a new house recently and I need advice on laying sod in my backyard.

The ground is rocky and clay like and likely not going to grow anything. However, they did lay a top soil all over the backyard. I'll have to follow up with them on how many inches of top soil they put.

The back of the yard goes upward like a hill and it has been raining, so I fear the top soil has thinned out up there. Plus, i see areas around the fence where it seems rocky and the original ground is visible.

Does it matter what type of top soil I get to add an extra layer throughout the entire backyard (or maybe just the areas where it's thinned out)? Or is it bad to mix different top soils and I should find out which they used and try to get that same one?

If it makes a difference I am planning to get Bermuda Celebration.