Well, "new
is glue" (Link). That means if you have new
'ed an instance, it is glued to your implementation. You cannot easily exchange it with a different implementation, for example a mock for testing. Like gluing together Lego bricks.
I you want to use proper dependency injection (using a container/framework or not) you need to structure your program in a way that you don't glue your components together, but instead inject them.
Every class is basically at hierarchy level 1 then. You need an instance of your logger? You inject it. You need an instance of a class that needs a logger? You inject it. You want to test your logging mechanism? Easy, you just inject something that conforms to your logger interface that logs into a list and the at the end of your test you can check your list and see if all the required logs are there. That is something you can automate (in contrast to using your normal logging mechanism and checking the logfiles by hand).
That means in the end, you don't really have a hierarchy, because every class you have just gets their dependencies injected and it will be the container/framework or your controlling code that determines what that means for the order of instantiation of objects.
As far as design patterns go, allow me an observation: even now, you don't need a singleton. Right now in your program, it would work if you had a plain global variable. But I guess you read that global variables are "bad". And design patterns are "good". And since you need a global variable and singleton delivers a global variable, why use the "bad", when you can use the "good" right? Well, the problem is, even with a singleton, the global variable is bad. It's a drawback of the pattern, a toad you have to swallow for the singleton logic to work. In your case, you don't need the singleton logic, but you like the taste of toads. So you created a singleton. Don't do that with design patterns. Read them very carefully and make sure you use them for the intended purpose, not because you like their side-effects or because it feels good to use a design pattern.