Slow upgrade from integrated - pre or power first?

I would go with the preamp. It's the next piece from your source before passing the signal to your amp. You always want to retain as much as possible of the original signal had work well for me. Good luck.
Nick

Power first then pre. If you look at the two sections of your integrated as separate components, the pre section (which is almost as good as a decent separate pre) when combined with a really good stand-alone amp will come pretty close sonically to a complete 'separates' system.

Whereas a new separate pre driving the amp section of the integrated won't really get to shine, and the result won't sound much better (if at all) than the integrated does now IMO.

And besides, if you can possibly try a few amps using the pre in the integrated, you'll likely be able to make a good amp choice. On the other hand, were you to try a number of different preamps driving the integrated's amp, the pros and cons might not be very obvious that way. Better to wait and audition preamps after you've made your amp selection. That would be my advice.

I had the same dilema a few years ago. A owned a well respected integrated. I was trying to upgrade one piece at a time and was auditioning a few amplifies. Quite by accident, I rotated a preamp into the chain and was shocked at the upgrade in sound. Secondly, my experience has demonstrated that a preamp is a more significant and impactful upgrade than an amp. I would experiment with upgrading to a stand-alone preamp first.
PS: I speak from my experiences with different equipment than yours so factor this into my comments.