I've been experimenting with techniques suggested here on a familiar song to try to work out what feels right and it seems to be thumb resting on either pickup or E string and just reaching across strings from there.
Interestingly, elbow position seems to be important for me. Letting it drop back and bending the wrist doesn't seem to work for me, keeping the wrist straight seems to help my hand move across the strings better. A long strap seems to help.