Next upgrade to an entry-level system?

A new phono stage is an improvement that would get lost in the resolving ability of the amp and speakers.  NOT recommended.  The Yamaha has a decent amount of power so I would consider the speakers the first upgrade, then the amp.  If you buy another, higher quality integrated it may come with an improved phono stage anyway.

Depending on the size of your room I would add a matching sub.  Two are better than one in my experience if you have the space.  

+1 on the outboard phono stage but I’d not get Schiit Mani unless that’s all you’re willing to spend.
Not sure what the budget is but look for a good used phono stage on a used market. Read reviews. Research feedback here. Come up with a list of candidates and repost. 

Get into room acoustics early

parkergetdean and ghdprentice are correct.

The best (and proper IMO) way to improve an entry level system is to upgrade speakers first. However, you want to upgrade them to a level that the rest of the system is obviously holding them back.....then as you upgrade everything else, they will sound even better. Buy used for sure as you upgrade

Works rather well in going from entry level, to low fi, to mid fi, to hi fi; I've upgraded several times on my way up over the last