Next upgrade to an entry-level system?

in my opinion, speakers are always the dominant influence on your system. So, my suggestion would be to start with the best speakers you can afford. Best meaning what sounds best to you, not to someone else. Its not a bad idea to consult forums or reviews to build a list of speakers to audition but take all that with a big grain of salt. Individual tastes differ, and you won’t find many reviews that are not influenced by advertising dollars, “accommodation pricing“ or some other conflict of interest. After you get a list, listen to many as you can and pick the one that sounds the best to you. Then you will have a good foundation that you can build around as you upgrade additional components as your budget permits

Good luck!

Echoing (sorry) Larsman's point, setup makes a huge difference. A good way to approach the task is with guidance. Jim Smith is the guru of audio expo setups. He installs and voices esoteric equipment playing in awful rooms, and his reputation for optimizing systems is stellar. His book is Get Better Sound, available as an ebook for $18.70 at the website of the same name. The book has 202 tips that cover the gamut. Art Dudley reviewed it favorably at www.stereophile.com/content/book-review-get-better-sound. It is likely the best use of your upgrade money.

I get no kickbacks, but my system is the envy of my musician friends.

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.