iPhone SE
So assuming you won’t consider the (excellent) iPhone SE and are trying to keep your new Android as close to five inches as possible, I’d steer you towards Google’s Pixel 4a (€390 directly from store.google.com).
It has just about the smallest screen you can get on a decent Android smartphone — 5.8in. Let me say a word about this, though. If you actually place this Pixel 4a back to back with your old HTC Desire 310 model, you won’t see much of an actual size difference. In fact, they’re exactly the same width, while the Google phone is only 12mm taller than the HTC, despite having a much bigger screen. The reason for this is that the technology has advanced to the point where new smartphones have almost no ‘bezels’ (the bits between the edge of the screen and the actual side of the phone) anymore. So the same size ‘phone’ in your hand now has a much larger screen simply because more of the front side is the actual display.