What Sasuke has is a complete Susano, not a perfect Susano since there is room for improvement. Neither Madara nor Sauce have proven to be capable of summoning the perfect susano like Edo Madara did, thus why my assumptions are that you require the Rinnegan's power to increase your EMS doujutsu powers. For instance, MS sasuke was barely capable of summoning an incomplete Susano, though once he got his EMS, he, without any difficulties, summoned an incomplete Susano, then later moved on to summon a complete Susano. You should have also noticed that his enton manipulation has improved.
I know this is late but, let me give a lesson.
I'll begin.
What Sasuke has is PS, whether it's complete or incomplete. Obviously right now he is using a modified version of it. Also perfect susanoo is just a name for it being in it's stabilized form and has no regard to the definition of perfect.
Also Madara and Sasuke both have showed the capability of summoning PS, when they cover Kurama in their modified PS. This will be my proof and reason:
If you accept that PS looks like Edo Madara's stabilized Susanoo (PS) then you have to assume Sasuke and Madara had the capability to use it. Obviously when they cover Kurama they are not using the full power of PS, instead they are only using a modified version of it. Since we can assume that they are holding back, that when they use their full power they can Summon PS in it's complete form.
Honestly the rinnegan has nothing to do with summoning susanoo or powering up EMS so you can use the PS.
Also Sasuke wasn't just barely capable in summoning the incomplete, he did it gradually over the fight. After he summoned it once he just kept progressing it all the way to final, after repeated uses of both incomplete and complete susanoo. Also he was in pain the whole time. Sasuke had already mastered summoning the complete susanoo with his MS.