You want to use this circuit as an opamp. Opams work with negative feedback. More specifically, this is a single ended opamp and you must have a feedback from the output to the negative input. Otherwise you can't even think that you will get a good operating point if you just simulate it open loop. So, the short answer to your initial question is - you can't guarantee that the second stage transistors, each one of them, will be in saturation unless you use feedback. You can actually guarantee with good certainty that if you don't use feedback, one of those output transistors will not be in saturation.
Something else I notice in your circuit - M6 and M7 are not sized quite well with respect to each other. It is good if you can make the ratio of the widths of M7 to M5 2x smaller than the ratio of the widths of M6 to M4. For that, it will be much better if the ratio of M6 to M4 is an even number rather than odd, as it is now.