Better Integrated Amp?

You didn't state a budget for speakers but I like the Vandersteen VLR Wood ($1495 USD). I have also heard it with the Proac D2 and Tablette's and both sounded really good. Being in the UK they might be easier to demo. The best sound I heard with it was the Vandersteen VLR CT but it's a bit pricey at $3100 US. If you have a local Rega dealer I would suggest that you go have a listen and see what the dealer recommends.

I am currently using it with Tannoy M1 bookshelf speakers. They are small and I have had them for 16 years. Maybe some current Tannoy's would be a good fit.

Good Luck!

Speakers depend a lot on listening preference, both volume and program. Also size/style desired. And of course budget.
For that TT the AT VM540ML is a match made in value heaven.

Has it been established that you are in the UK???

Get the most powerful one you can afford.
https://www.exposurehifi.com/products.html

This Yamaha A-S501 is worth looking into. It has all the features that you will ever need!
https://usa.yamaha.com/products/audio_visual/hifi_components/a-s501/index.html

I would focus on auditioning speakers 1st, and consider the amps that are paired with them during audition. 

What budget are you considering for speakers?