Other things to check:
Is the low gear stop adjusted a bit too tight, so bottom gear isn't fully clicked in at the shifter?
Is the shifter badly worn?
In the unlikely event that you're using a friction shifter, frame flex can tug the cable enough to move the lever*. This shouldn't happen with indexed gears unless the shifter is really worn out.
You can check hanger alignment with a long steel rule and calipers, but you should remove the BB, wheel, and mech first. This method also assumes the BB shell is in straight. You should check for inward/outward bend and twist. I have an old Cinder Cone MTB frame that had a visibly bent hanger, but after 20 minutes' careful tweaking it shifts impeccably.
*my old Thorn would always change to the middle chainring if I tried to sprint in the big ring