With all-star polar alignment and some experience, you can routinely achieve polar alignments well within ten arc minutes of the celestial pole. With the functionality of modern mounts and autoguiders, this is often sufficient even for demanding astrophotography.
The classical declination drift method of polar alignment is still the gold standard. If you really want to take the additional time that the drift method requires, you can combine it with all-star polar alignment to hone your alignment. There are two basic ways to do this.
Use it before turning on the mount to mechanically align the polar axis. Then turn on the mount. The polar alignment should be pretty close and you can hone it even further with the electronic functions of the mount. Follow through with realignment on the sky and calibration stars for the best results.
An alternative would be after doing the electronic polar alignment with the mount, then power down and perform the drift method. Doing it this way, the mount would already be mechanically very close and not as much adjustment would be needed. You would still have to align the mount to the sky (not polar align) after turning the mount back on.
Either way you choose to do it, never leave out the calibration stars, as they tell the mount other things besides alignment and polar alignment (perpendicular axes, etc.)