I leave the feet for the end. One idea is you could wrap the foot in a hot towel and massage through it. My clients love having hot towels on their feet. There are a lot of great ideas listed here!

I always did the face and head first, while my hands were clean...most people aren't that gross that it's a worry to touch their feet, then other parts (not face) of the body?  Do you just have feet issues?  I wouldnt' like hand sanitizer on my feet (chemicals in that stuff are bad for me, and YOU who might be touching more often), peppermint foot lotion might be a nice addition...essential oils are anti fungal, bacterial and viral...

These are all great ideas! As a reflexologist and former massage therapist, here are my thoughts:

(1) Use a hot towel to wipe off the feet at the beginning of the session. That way they are clean whenever you get to them and you won't have to break continuity. Contrary to popular belief...feet are not the dirtiest part of the body (hands and face are dirtier!) because they are usually in shoes. An antibacterial wipe (use a baby wipe warmer to keep warm) is another option. 

(2) Feet that smell are giving you important information about the body. Typically, these clients could use more relaxation and lymphatic work.

(3) If the feet REALLY smell and/or have signs of athlete's foot, etc. do like Qiana suggested and work through the sheet.